This article from early this year explains how to apply in Plymouth.
https://www.plymouthherald.co.uk/news/plymouth-news/how-apply-sought-after-social-3709082
And this from local government...
https://www.plymouth.gov.uk/housing/findhome/applysocialshelteredorhousingassociationhousing
I realise you may have seen these already.
I've lived in Plymouth for 20 years, and must say I was severely disappointed in the look, and feel of the city. It felt small, ugly and not very cosmopolitan.
It has changed a lot over time, partly due to the growth of the university and investment by the city. There has been much new construction along with the transformation of many old military buildings into striking apartments. Lots of new and varied restaurants, a new museum complex, a huge entertainment hub. With a car, Dartmoor or the sea are both just a short drive away. Shopping is still a bit rubbish though, and there's still areas that look tired.