Can I send my MOTO X(1sT) to some place to change the battery?
Perhaps, however, if I were determined to hold onto a now going on six year old phone, I’d take it to these folks: https://www.cellphonerepair.com/.
There are a few issues with wanting to repair the 1st gen moto x. I repaired mine few years back and had to try out three different batteries from different sellers because the batteries are old and most of them have gone bad regardless of where they are stored. Luckily I found a new battery that worked perfectly. Chances of you finding one? Low. Most places would probably refuse to repair the phone. You can try your luck on getting a newer compatible (with older republic plan) phone from ebay and repair that one if you need to. That might be the better option. I would have tried something like that. Another option is you can try contacting motorola directly and pay their repair fees if they agree to fix it.