I would say try replacing the shift cable, it's very cheap.
On the other hand the rear derraileur sits at the smallest cog when no tension is applied at the wire. So for some reason your shifter (not a fan of the word brifter) winds but doesn't release.
Another approach is to WD-40 the heck out of the shifter mechanism instead of the chain and derraileur, maybe there's something sticking inside.