Most overcoats that are wool have the thick wool as the main part of the coat and then a satin liner inside and that's it. They are not typically "stuffed" with insulation such as a puffy winter coat might be.
I'm your size as well, with similar health issues, and I find boys size 18r work great for shoulder fit, etc. Much cheaper than men's 34s and often the same quality from major makers like Brooks Brothers. I also find Brooks Brothers ESF or SF 14.5/32 work great. Steve Smith here on the forum resells Brooks Brothers items and I've purchased a ton of shirts from him in ESF/SF.