Scarlet. That sounds tough. Have you read Hugh Mariotts 'Selfish Pigs Guide to Caring'. If not, get a copy as soon as possible. It describes the emotions people dont feel able to talk about which helps.
Worth reading also is 'Contented Dementia' by Oliver James. I was not convinced by the approach. However the chapter on when is the right time for a care home seemed useful. Struggling on your own may not be the best thing for either of you.
And a thought for Waterlego. By chance when my mother first had a fall I did some desk research on the cost of convalescent care by Googling local care homes and phoning them. The hospital discharged her early and unexpectedly at a point when we were only really picking up on her memory problem and had little understanding of the implications.
If money is not an issue it might be an idea to line up a Plan B in the form of a home who might take your dad at short notice for a temporary stay. This then gives you a bit of breathing space if you need it.
Where my mum went for 5 weeks was utterly luxurious and she was sufficiently confused to think she was in a hotel. It meant I could get on with practicle things like sorting more suitable permanent accommodation and trying to make sence of her finances, knowing she was safe, well fed and being entertained. It cost an arm and a leg but worth it.
Do the POA as soon as poss. Better done before a diagnosis, and it sounds as if your father is getting worse fast. (Though you might well find this is down to a UTI or similar.) In any event it takes about 3 months to come through. However much longer if you have to go through the courts. One thing you could do quickly is get forms from the bank which allow your parents to add you as a signatory to their bank accounts. Then you can at least access some of their money should both be incacitated.
We got the priest to be witness to the POA. My mother was very nervous of signing a POA but also clearly rattled by her fall. She trusted him and he was able to reassure her that this was something she needed to do.
For me it all kicked off around Christmas. There was a lot to learn in a short time.