Yes, squeezing out the sphagnum moss is the right way to do itI would say that if he still has stuck shed tomorrow yes it's a good idea to give him some help. We have some great threads on this forum showing how to do it if you need any guidance.
I used to have a similar problem with the cool side of my enclosure. I have a 30g Sterilite tub and the ambient temp in my house in the winter is around 70F. My hot spot was at 90F with a UTH but the surface of the substrate on the cool side would fall to 73F. I ended up fixing the problem by installing a CHE on a separate thermostat on the cool side and setting it to 80F. The ambient temp is now 80F on the cool side and 79F on the warm side. That's a solution you could consider if you're having trouble with ambient temps and the cool side being too cold.