Have you double-checked with your GP just to be sure it isn’t something simple like oral thrush? It can be painful and cause all sorts of issues if it gets out of hand in someone with a suppressed immune system. I realise you probably have, but just in case.
Certainly I get similar to what you are describing, but the dermatologist seemed to think it was connected to Crohn’s, not psoriasis. My symptoms fit those you’ll find if you Google orofacial granulomatosis, or oral Crohns. My lips swell up and when they return to normal have palpable lumps in them, as well as the fissures and sores.
A number of autoimmune conditions are associated with apthous ulcers as well, I had a lot of these at onset of PsA. Dry eye also includes dry mouth and will cause problems too.
The only thing I’ve found so far that works to resolve symptoms is mouthwash made of steroids, or systemic steroids, though I haven’t been on a biologic whilst these symptoms are happening, and I’m finally starting Remicade in 3 weeks so fingers crossed that will work.
When it is bad I use the over the counter anaesthetic mouthwash before eating and just before going to bed, and if it wakes me up at night.
I think it might be really worth while going to see an oral specialist though if you are able to?