Did it for a summer. Helped a buddy, now and then, for a couple of years.
Always slung the double bags over the handlebars. Big, wide "cruiser" bars and ultra-long wheelbase, so not all that unstable, all things considered.
Was on seriously flat terrain, on streets with sidewalks and bike lanes, so not a bad place to have a paper route. Made a little money. Got to know a lot of the neighbors in the area, back when hanging out at the mailbox chatting with Mr. Johnson (or whomever) was quite the thing to do, in the morning. Don't ever recall getting jilted on the monthly subscription fees, either.
Edith was right. "Those were the days!"