MacBook Air 2013 turns off without warning but only on battery power

I am experiencing an issue with a MacBook Air from Mid-2013. I should preface that this unit was used in Thailand and Sub-Saharan Africa by a student/peace corps volunteer over the last 4 years. As such, it has experienced a lot of damp heat followed by a long period of very hot, dry working environments with a lot of dirt and dust. I was cleaning the exterior of the unit when I was informed that the unit powers off randomly when on battery power. I tested it thoroughly on AC Adapter power, and while the magsafe connector didn't have a firm connection (probably need to go back and clean the ports next), it still supplied power, and I saw no issues.

When on battery power, however, the device is very temperamental. It boots up without a problem, but it always turns off at some point between 1-5 minutes of starting up. There is no warning for this, the system simply loses all power (screen goes black, fans turn off, etc.) I cannot guarantee that the battery is still healthy, but the computer does not report any battery problems, it appears to charge fine, and reports the battery as being at around 90-100% power when the shutoffs occur.

No physical stimuli seems to correspond to shutdowns. It will happen whether the unit is stationary or if it is moving. There appears no correlation between specific programs running, as it happens regardless of what is running, including nothing running at all.

I am not detecting any heating issues. I have also reset the SMC.

My current theory is that something has affected the connection of the battery to the rest of the machine. I'm on the fence as to whether I should replace the battery outright. I'm also considering doing a deeper cleaning of the inside of the system since it probably has a good amount of dirt inside of it.

Thanks for any advice on this matter.