Should I share a cancer diagnosis?

Lets's not jump ahead. That is a lot to assume. The fact is that of all the things that could be wrong here, Cancer is the most unlikely of them. Swallow problems are common in many dementias and just in aging.
What is needed here is a good swallow eval by OT. Contact the doctor.
As for myself, as a nurse, I believe hiding the truth from a patient, from ANY patient, is a terrible mistake.
Please don't be afraid to discuss the "dying" remark with Dad. Ask him for details about why he feels this way. Ask him if it is his heart problems, or his swallow that is of concern. The fact is that with CHF your father has a weakening heart. You know him best and what he wants to hear most, but denying what he is going through may NOT be reassuring.
I would say "Dad, why do you think you are dying? What is worrying you? You have some CHF and that means your heart isn't as good a pump as it was when you were young, so handling fluids in your lungs and in your legs might get to be a problem. You wouldn't expect your heart to be the same as when you were 29 years old, right? What bothers and worries you the most right now? Can you talk to me about it? Is there something we can do? Because we can see the doc about things of concern and talk about them."
Some people are more afraid of death than others. It is a lot for anyone to absorb, right? But he knows where he is in living. Let him talk with you about all this.
I had patients who waited until their family left the room to speak the truth with ME, and that's kind of sad.

As others have noted, first he would have to be diagnosed with cancer. IF he is, let me share my experience. I was a home health RN assigned to admit a man with a diagnosis of untreatable cancer. All over the admission paperwork was written "patient does not know, family does not want him told". Yikes. My first couple of visits, the wife hovered over me I felt was to be sure I didn't say anything. After developing a relationship, I encouraged her to go out for a walk while I was there. On that day, he turned to me and said, "I know I have cancer, they just don't want to talk about it". Trust me, patients know. Then he turned to me and said, "Am I going to die"? I knew what answer he wanted but told him, "yes". As he startled, I told him we ALL die. The difference is that none of us know when or usually how. As I shared with him I could be hit by a bus leaving him and die. It was important to live every moment and time for the family to open up together. He agreed. Next visit I said to the wife, "he knows he has cancer, you know. HE told me. Now it's time for you both to share and love and open up." They did and found peace and acceptance and of course sadness.
IF he gets a cancer diagnosis that is deemed terminal, get Hospice in ASAP. They are NOT about giving up but living every day to the fullest. The staff are well experienced in how to help those with the fear of dying. Do get them in early and not wait. You do NOT need a doctor's order or permission to have them do a home visit for information.
Best of all peace and golden light to you all.

I don't think I am understanding why you have automatically jumped to the conclusion that it must be cancer. He might just be having trouble swallowing his food, and that is causing him to gag. You can take him to the Dr where they can perform a swallowing test to see if in fact that is the issue. In the mean time, I would only give him really soft or pureed foods, and drinks that are thickened. My husband ended up with aspiration pneumonia, after he starting gagging on his food, and I was told that it was caused by his vascular dementia. That his brain wasn't communicating any more with his throat, telling it to close as needed. He almost died because of that, and I had to keep him on thickened drinks and soft foods for the rest of his life. Wishing you and your father the best.

My mother had CHF, as they increased her diuretics she had trouble swallowing food; everything was very dry to her, even things I cooked in a crock pot.
Her cardiologist told me diuretics can cause dry mouth, which can lead to trouble swallowing.
I agree with calling his doctor, there might be something they can give him to mitigate the symptoms.

Why expect Zebras? (old saying when searching for a diagnosis, If you hear hoofbeats think horses not zebras)
Gagging or choking can be caused by any number of things.
You do not mention anything in your profile about his medical history.
The choking or gagging could be because he is not swallowing properly, it could be because of acid, it could be inflamed tonsils, ....
Make an appointment with his doctor, if he was seen by a Gastroenterologist make the appointment with that doctor since they would have previous information about his health.
IF your dad has not been diagnosed with any form of Dementia then YES the diagnosis should be shared with him and he should be involved in his care plan. IF he has dementia then there would be no need to share the diagnosis IF you do not plan on treating it and just contact Hospice.

My dad had a similar problem, and the doctor gave him nitroglycerin tablets to use when it happened. He was just having spasms in his esophagus. The pills cured him.

Easy peasy.

Take him to the doctor.

I think I'd weigh up the pros and cons of sharing a cancer diagnosis after my loved one had actually been given one. Aren't you rather crossing a bridge before you've come to it?

What investigations are being ordered?

Hi, I used to take care a 95 woman who just passed away a month ago.. she was always telling me that she was scared and didn't want to die... so what I used to tell her was this:
"We're all going to die one day, so we should be grateful for everything we already had during our lives and try to find peace, prayer meditation can help to relax. " Also think that death is a release of all our problems in this world"
I think that the place all the good people go when they die is a wonderful and happy place because nobody has come back from there. 💞 Much love and Blessings 🛐

JulieAnn,
Hoping you will have your father tested. And get the advice of a physician.
People who decide to get palliative care or forego treatment are first dealing
with the facts, using medical science instead of relying on our own feelings about another's health. This is not said to hurt your feelings. Of course you are worried for your father. Jumping to conclusions will make it worse for you and him. imo.

And I do not want you to create more worry for yourself. You said:  "I just have a bad feeling about this. If the problem turns out to be cancer I don't see any value in telling him." That is not how cancer is diagnosed. And your father has rights.

It is much too soon to be considering these things. I know this because my loved one was told "it may be cancer" and his daughter went around crying wolf for the next 6 years: "My father is dying of cancer". He did not have cancer.

Funkygrandma gave another possibility. Just wait up, more answers will come, and I hope people are kind and understanding to you. Now pull yourself together, wash your face, and take care of business, despite the challenges.
💞🍵💞

He was clear with you that he's afraid of dying, so I wouldn't share the info. A cancer diagnosis, if you've never had one, is total panic to begin with and it occupies every minute of your time with thoughts about it. That's just at diagnosis and while you're waiting to get an appointment for the next phase.

To back it up a bit, tell dr prior to first consult that dad is very afraid of dying so if he wants any tests done - then leave the word cancer out of it for now. If dr thinks it could be cancer or wants to rule it out, they set you up with appt that won't happen for a week or so. THAT period of time starts the whirlwind of thoughts, so no point in being a nervous wreck before the test. You finally get to the appointment for xray, mri, etc and even if you ask what the tech thinks, they won't tell you. So now you play the waiting game again for someone to review test and another appointment with the referring doctor. I would NOT tell him why he is having tests. There's always the chance it's not cancer, so leave it be.
If it turns out to be cancer, you can have the initial conversation with the doctor to find out what a treatment plan would be and how hard it would be on him - or if dr even recommends treatment considering other health issues. That's when you can talk to dad to find out how he wants to proceed. It's no necessary to use the word cancer in the discussion either. You could say, there is a growth near your 'whatever' and doctor thinks some meds (leave out chemo) would help. (Actually some chemo is giving by pill, not )always IV. Or doctor thinks a little radiation would help.