Once I started thinking of the lycra shorts as athletic gear, I found I didn't care anymore. I need the support on the bike just like I'd need swimming trunks, golf shoes, or shin guards (for soccer). If I felt the need to take off bowling shoes before getting some nachos, then putting the shoes back on before bowling the next frame, I might feel the need to change shorts before popping into a store or gas station to get a snack or drink; but I don't, in either case. Most of the people I run into in those establishments don't seem to care if I don't care.
Two things might help. First, I normally wore some kind of high-visibility jersey on tour -- that eye-searing green, yellow, or orange attracts the eyes of the clerks and other customers. If you're still shy, I tried to get out of the shorts as quickly as possible after the day's ride for crotch hygiene. That means most of the people you deal with in a day, at a campground, restaurant, library, motel, or grocery store, will see you in looser clothing.