Is my MacBook Air worth installing a new battery, or should I purchase another computer?

This is really a personal choice you'll have to make.

For me, 10 years is a VERY long time. I typically upgrade every 4-7 years or so.

You are already past the point of MacOS updates, as you most likely already know. You might want to take a look at the current MacOS, Ventura, and see if any of its features are worth it to you. macOS Ventura - Apple

Additionally, you'd gain the benefit of a much, much faster computer compared to what you have now. More and more apps are being released that truly take advantage of the Apple Silicon architecture, and you'd be amazed at how fast they are.

But I also know nobody has unlimited finances, so I can truly appreciate why you might want to hang on to your current Mac.

It's an obsolete, no longer supported, device according to Apple. Pretty much anything you do to it will be of greater cost than the worth of the Mac. Your call ...

Joelspin, you're welcome. Hopefully we'll see you soon here with a shiny new MacBook Air.