Shammeeee for the cloth and Sham mwoa for the deer thing.
I do this. I don’t know why as I know the first one is wrong, but it’s just what people call the leather rag things, and I’m happy to go with the crowd.
There’s no rhyme or reason when it comes to using a foreign pronunciation.
Look at Chamonix and Paris. Pronounce them in a totally English way and Chamonix sounds ridiculous with the Ch for Chariot and X for X-ray substituted in. Go full frog, and get a punch in the nose every time you say Paree.
I do pronounce Moet with the T, purely because I found you should, and it’s a chance to be a smarmy contrarian know it all.