How do you manage fecal incontinence?

She needs to see her Dr and have a rectal examination that will show if she is full of hard stool and liquid is just dribbling out around it. She can be given an enema or disimpacted, THEN manage the diet and laxatives as advised.
It is not especially unhealthy to sit on the toilet for long periods as long as she does not pinch the nerves in her upper thighs and be unable to stand. You can buy inflatable toilet seats which don't give pressure

Limit her time on the toilet, if nothing is happening after 15 minutes have her come back to try later. What you are describing makes me think she had hard stool that isn't being passed, I would start by doing what it takes to ensure her bowel movements are fairly soft - that might mean adding whole grains, fruits and vegetables to her diet and making sure she is well hydrated, it might mean adding in an occasional laxative or she may need something more than occasionally (the forum is full of advice about constipation and everyone has their own needs/preferences). Of course she should be using a disposable pull up as well.
Unfortunately some people with dementia can become fixated on their BM's and are creating problems where there really are none, that could be part of your problem as well.

You may need to let her sit for a while. You are going to think this is weird but my Mom was told this. Her bladder had dropped. She was told to push on the area that episotomies are done while you push. It really helps release what is there.