Does a full guardianship have the authority to place a loved one in a nursing home?

IF mom's MD recommends placement and the facility evaluates and accepts the patient, you should not have a problem. Be sure you notify the court and the conservator in writing, with the supporting letters of recommendation written by the MD and the facility.

There is enough money for a couple of years,after that then will have to try and get mom on medicaid

Oh, that does make a difference. You are "guardian of the body," right? You and the conservator would need to work together to pay the NH. As long as you and the county are in agreement and there is enough money, there shouldn't be a problem. Do you think that you may need to apply for Medicaid in the future? or will there be money to cover the NH for a long time?

I am full guardian, sheriff is over moms finances. Mom is the last stage of Alzheimer's

A guardian can do what is in the best interest of his/her charge. If that is placing them in a NH to receive the care they need, then yes, a guardian can do this.

I checked one of your previous posts and learned that you apparently hold guardianship but there was some issue with a sheriff holding conservatorship. Is this still the case?

If so, and there is a split between the responsibilities, the order appointing you guardian is the guiding document for what you can and can't do, and the conservatorship order also establishes what the sheriff's department can to.

Is that responsibility granted in the order of appointment for you as guardian?

Are you asking because you are the guardian, or because you are not?

It really depends upon the state and the court, but my understanding of most guardianship is that it's like being the parent of an underage child. You get to say where s/he lives, goes to school, etc., but there are regulations that require you to keep the person safe.