Hi all, I have an update: I have FINALLY found an app that does this!
It's called Sellbrite for Shopify (not to be confused with just Sellbrite). You can link 1, multiple, or all of your Shopify locations to Amazon, Ebay, etc. You can even set up multiple Amazon accounts, linked to different Shopify locations if you want. This means that inventory is shared only between the location(s) you want, not necessarily all.
You can create the listings directly in Sellbrite, but you can also make listings in Amazon directly and they link automatically by SKU (you can even unlink them if you want, though I'm not sure why you would). We prefer to make Amazon listings directly, and also set up pricing directly in Amazon -- you can make pricing sync to if you want, but we do not want to.
Shopify remains the source of truth for inventory and all orders get into Shopify, meaning you can do your normally Shopify fulfilment (ex. with Shopify shipping), without ever going into Sellbrite after the initial set-up and occasional checks that everything looks fine.
Best of all: it's free up to 30 Amazon orders per month, and then $19. So way cheaper than any of the other apps I found, which were way to complicated for my needs.
All in all, I am really happy with this app and wish I had found it sooner... maybe someone else can benefit from my months of searching