Good advice. Everyone's circumstances are different and we all have different tolerance for risk. A reliable elder attorney is a good start to find out what is available to you and what measures you need to take in your state. Whether you go with conservatorship, a family trust, or some other legal tool, it needs to be legally set up and the family should be aware of it. Powers of attorney were/are invaluable to me in dealing with my FIL's finances and medical care. I have both durable and medical POAs.
Because your FIL has a girlfriend, you are not only dealing with your own family members but hers as well. Without properly setting things up and watching like a hawk, her family could end up with all his assets.
Money is a seductive thing. As Jim says, it's your FIL's money. But, when you assume responsibility for managing it, it can seduce you into thinking it's yours. There is a reason nice families fight nasty over inheritance money. Just keep reminding yourself that it is his money, it is his money, it is his money.
I don't know how involved you already are with his finances but you may be in for a shock. Do you know for sure whether there are any second/third mortgages? Liens? Reverse mortgage? What does he have in the bank? Under the bed? Tucked into picture frames?
For your own protection, you need to know these things and be able to at least look at his bank accounts on a regular basis. I check my FIL's accounts on-line daily--and I clearly let my SIL know I am watching.
I started about 8 years ago--long before dementia was a diagnosis--to set up legal tools for my FIL. At the time, I wasn't even thinking he would become unable to manage on his own. I was watching out for a BIL with mental illness issues. I knew if he got a windfall, his "friends" would relieve him of that burden in less than a week. Setting up a trust to dole out said BIL's inheritance a little at a time was my goal. The lawyer also set up the standard POAs, etc. Little did I know how useful those would become.