Thanks for all the great suggestions I did not but the Hollywood, but I did end up buying a vintage (1950s) Schwinn Starlet for $60. I cannot believe I am riding a bike that is 30 years older than I am, but it feels like it was made for me. Coaster brake, just like I wanted.
Took it out for 2 hours on the bike path this weekend, and it was just lovely. Lot of stares though. It's got the whitewall balloon tires and the tank horn, so it's a bit more of an eyecatcher than I would prefer.
I found a pdf file of the original catalog from the 50's. Turns out it's actually a girls"youth" bike. It feels sturdy though, and I'm actually no taller than the average 10 year old, so I think it's alright.
It seems like a pretty simple bike, but at this point I am not comfortable trying to do any work on it myself. I'm probably going to ride it through the summer, and take it to the bike shop closer to winter, sooner if I run into problems.
For $60, I'm not going to complain.