Great idea! The problem will be finding a furnished house to rent.
We wanted to do this and stay for a year, but could not find a furnished long term rental. Instead we did longer trips, spending 2 weeks in a few different locations, because that was easier to book than in one place. This was about 15 years ago.
One winter we found a holiday rental that gave us a good price for a 3 month stay, January to March. We had friends in the area who went to look at the cottage for us. We had a good stay and looked for longer term rentals while we were there but found none furnished.
Eventually, a couple of years later, we were able to rent that same place for a year, and we came to the UK with our cat. The house was in Nailsworth, in the southern Cotswolds.
We ended up staying in the UK and are still here 11 years later.
So, my advice is to choose 4 locations and find holiday rentals for a month in each place. Ask for a discount because you are staying a longer time. While researching that you may come across a place you could rent for the whole time. Most holiday rentals will have some weeks booked already for next year. Prices zoom up for school holidays - mid July to end of August.
@artnbarb may have something to add because she considered something similar and has done longer stays.
For locations, there are so many great places to stay here. We spent 5 years in the Cotswolds but now live on the Dorset coast. I love the southwest but there are other great areas. Where have you stayed before, other than Dartmouth?
Many holiday rentals take dogs, so you are good there. I recommend holiday rentals instead of long term rentals because they are furnished, you don’t have to sign up for utilities or pay council tax.
You can stay in the UK for up to 6 months as a visitor.
Every small town has a NHS office. Our town is population 12,000 and has a good NHS. Only villages won’t have one but there will be one in a nearby town. You probably need your own health insurance. I’m not sure how all that works, if you can use the NHS as a visitor.
Good luck with your search!