We mainly solve this by not wearing nicer shirts! 😀
But seeing as that's not particularly helpful...there are giant hangers with wide shoulder arms, my husband has some leftover from suit jackets that he uses on the rare occasion a dress shirt has to be washed. That avoids those weird dimples. There are also sweater racks where you lay the item out flat on a mesh type surface for air circulation. We don't have one but I suspect it would be pretty easy to diy one.
We went to Berlin in winter one year for vacation. The apartment had a tiny washer and a drying rack. That experience totally kicked our butts back into drying everything on a rack! Good luck!