Andy, I think some musicians (like you) are happiest playing from music, others learn tunes by ear. In my day we were taught only to 'read', not to hear, I'm glad that is not so now and both skills are valued.
So sessions are great for those who naturally pick up tunes by hearing them.
Not all sessions completely ban 'the dots'. Some will even hand round music for an occasional new tune.
I found the session that helped my learn to play by ear was a mixed session in the Shroppie Fly. Some helpful singers who shouted 'key of G' and all the instrumentalists noodled along.
It's not quite the same as 'learning a tune by heart', it's more teaching your fingers to play the intervals of a tune that's in your head without needing to see it on paper. Then maybe learning the difficult parts of the tune by heart.