It does sound like you have a lot going on with your poor Pixel. As a fellow Pixel owner, I can sympathize.
Okay, first things first, genuine Google replacement screens are NOT hard to get; you can order one directly from iFixit! They have an agreement with Google and sell only genuine parts for Pixel phones. Here's a link for yours.
Google Pixel 6a Screen - Genuine
I would suggest the Fix Kit version, as for an extra $7 USD you get all of the tools and adhesives needed to do the job. Here's the step by step guide showing you how to replace the screen.
Google Pixel 6a Screen Replacement - iFixit Repair Guide
If your screen is already starting to separate, that makes it that much easier to remove it since the hardest part can be getting started on separating the screen from the frame. However, that in itself may be cause for concern. While it is possible that some of the dings and bruises your phone has experienced could be the cause, by far the most common reason for a screen to be separating is if the battery is swelling. That happens when the chemical reactions in the battery have started breaking down, generating gas and causing the outer casing on the phone to swell up. In that event the battery MUST be replaced, so you'll want to check that when opening the phone. If the outer case is flat against the battery and it's all square and everything you're probably fine, but if it looks pillowy and has give to it, then a new battery is in order. You can get that here as well.
Google Pixel 6a Battery - Genuine
And of course you'll need a guide to that repair as well.
Google Pixel 6a Battery Replacement - iFixit Repair Guide
Note that if you find you need a battery and a screen, you only need to follow the battery replacement guide, since removing the screen is the first step to getting to the battery.
I will say that in my opinion I'd say your screen has probably been damaged by the liquid exposure and it's highly likely you're going to need to replace it. Of course I only have what you've told us to go on to base that opinion on, but that would be my best guess.
In the future, please don't put your phone in rice. Experiments have actually been done to find out if that works, and the conclusion was that it doesn't. In fact, it can cause problems due to dust and rice getting in places it shouldn't be like the charging port, speakers and microphones. It's a good theory, but in practice it's not of any value. If you want to do something like that, you could buy a bag of silica gel packs (or save them up in an airtight container) and put your phone in that if it gets wet again.
You certainly could try opening up the phone and cleaning the connector with 90% or higher isopropyl alcohol. While it's unlikely, on your phone the display connector is only held on by a spring rather than being screwed down like it is on my Pixel 6 Pro. So it's theoretically possible the retainer has popped out and allows the connector to come loose, but as I said, there's a very low chance that happened.
While the OLED display may be working just fine, the problem is it's bonded to the digitizer (touch screen) and requires specialized equipment to separate and rejoin, so for us home DIYers it's just not practical to try and replace only the digitizer; you really have to get a whole new screen.
Mark's warning is valid regarding the delicacy of the screens, but with care they can be safely removed. The flip side of that is if you're replacing it anyway then it doesn't matter if you damage the old screen. He has, however, missed the whole point of this site, which is to empower you to be able to repair your own devices rather than take them to a repair shop. If you don't have a lot of experience with phone repairs you could take it to a shop for a diagnosis; they should be able to tell you for sure if you need a new screen and/or battery - they can probably plug in a new screen and verify it works correctly. They shouldn't charge too much to do that, and then you'll know what needs to be done and can do that yourself.
Oh and finally, if you find the need to replace both the screen and the battery, I'd suggest getting the Fix Kit version of the battery and just the part alone for the screen. The battery fix kit will have everything needed for the screen along with what you need for the battery, so no need to buy stuff twice.
I hope this gives you all the information you need to get your Pixel back up and running. Good luck and be sure to come on back and let us know how it all turns out! Your experience may well help someone else who runs into the same situation in the future.
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