You asked "Suggestions for how to add storage shelves to a utility room coat closet without major cash outlay?"
I bought a set of put together shelves like the one in the link below and put it into a hall closet, works great. These type shelves just slip together with no tools needed and come apart just as easily. They also come in different sizes and I think sterilite makes some as well. They are also available at different stores....Amazon carries a wide variety of storage shelves too.
Hope this helps and you get better soon.
https://www.amazon.com/s?k=storage+s...f=nb_sb_noss_1
https://www.walmart.com/ip/Plano-32-...-Taupe/8251818