Hi Micah,
Okay, I'm going to start out by making an assumption; please tell me if I'm wrong. If you simply followed the iFixit screen replacement guide for your phone, then you wouldn't have had any need to mess with either the battery connection or the battery itself, right?
That being said, it would appear there is more internal damage than just the screen in this case. Specifically, the evidence so far points to either the battery or the charging port, given the failure to charge.
Unfortunately, Google made the decision to change from the charging port being on a separate circuit board connected via flex cable, to mounting the USB-C connector directly to the logic board, making charging port replacement an order of magnitude more difficult. On previous models you could simply order the charging port assembly and replace it in whole, while on yours you're looking at a soldering job to get the old one off and replace it with a new one - assuming you can even find the correct part (I couldn't). Visually, it appears to be identical to the port used on 5, 5a and 6, but I have no definitive information to say whether it will work or not. Here's a 5/5a/6 port for sale on AliExpress.
Original For Google Pixel 6 Pro 6pro 5 5a Usb Charger Dock Charging Port Replacement Repair Parts - Mobile Phone Flex Cables - AliExpress
Vs. the port as it appears on the motherboard replacement guide.
So at this point, if it was my phone, I'd probably be looking seriously at replacing the battery. The battery shouldn't be prohibitively expensive and the repair is fairly straight-forward. Plus, of course, if it's not the battery, then the difficulty of repairing a charging issue goes up exponentially - if it's not the battery or the charging port itself, then you're looking at board-level microsoldering repairs that can get very expensive; sometimes to the point where it's cheaper to just buy a new phone.
You can buy the battery right here on iFixit; I'd recommend the Fix Kit version; for an extra $7 USD you get all the tools needed to do the job right, plus all the adhesives needed to put the phone back together correctly.
Google Pixel 6a Battery - Genuine
And, of course, the guide to do the job is handy to have.
Google Pixel 6a Battery Replacement - iFixit Repair Guide
Let us know how it goes, and best of luck with your repair!