@sprigatito I've come to this thread today and read all of your posts but not the replies, so apologies if these things have been mentioned before.
Firstly, you are doing an amazing job, never let yourself forget that. I was a community carer for dementia and alzheimers clients, as well as those under home hospice care, and it is tough, that's without close relative level emotions being thrown in.
Are you registered as a family carer? If not, do it. Google the name of your county followed by "family carers" and you should find a link. For example, ours is called Suffolk Family Carers, North Lincolnshire is called The Carers Support Service, but whatever it is called you'll find it by searching for 'County' family carers. Register with them. It doesn't matter that dad officially lives elsewhere, you are still a family carer and can access information and support through your own county. They can do a carers assessment, they can fund things to help you in your caring role. For some people that's a holiday, for some it's a day centre placement for the person they care for, for you I'd suggest asking for a railcard and transport vouchers to make it so you don't have the cost of travelling back with dad to his home or to his appointments.
You mentioned not being able to leave for the walk in centre until you'd persuaded dad to wash and change his clothes. To put it simply, Don't. I know it goes against every instinct you have, but the people you are taking him to see will not be bothered and will not judge you, him, or your ability to care for him because he hasn't washed and has clothes on that are not perfectly clean. Take him in his pjs if you have to! Keeping your stress levels down for the sake of your mental health, whilst still getting dad to the right people to help him is far more important than him being clean and well presented. If you're going to get through this you will need to adopt a don't sweat the small stuff approach. Focus on the big goals like getting him there rather than the small ones like his clothes.
Consider telling the doctors etc that you take him to about you being ND. Tell them you need to record the meeting on your phone because it helps you to listen again a few times as you take longer to process what is being said than an NT. Ask them to write down the important parts of what they're telling you. Just because the treatment is for your dad doesn't mean that you aren't entitled to those reasonable adjustments. It will also potentially flick a light switch for them to think that dad may also be ND, although undiagnosed. The better ones will adapt their approach as they recognise the statistical probability that he is too!
If you need a break from the family messages etc, take it. Tell them you are overwhelmed and you will contact them when you are able to but not to contact you first. If you don't feel up to being that honest with them just mute your phone and keep it in sight so if it is someone important you can answer, or give the relatives a different ringtone so you know you can ignore that call/text, or ask DH to keep hold of it when he's around and only alert you if it's doctor or hospital.
Look up Attendance Allowance. Dad may well qualify and that will help financially. If he isn't already on PIP then look up that too, if he is then you should apply for Carers Allowance for you. These things will help you to help him.
Lastly, take a breath. It is so difficult to prioritise yourself and you likely feel bad if you do. You can't pour from an empty cup is used in the family carers world a lot - if you are exhausted and drained you won't have enough left to give dad what he needs.
I'll be thinking of you both, you've got this ❤