I know there are plenty of tech savvy folks on the forum, so I'd like to hear your thoughts. I'm not going to change to Linux just to keep an ailing machine, but I can't see replacing a machine which still functions properly.
Honestly, why not?
Mint or Ubuntu. The issue of OS replacement is a difficult one--it's like if you re-installed with Windows 7. Windows has the upgrade path advantage, but you've waived that.
But I use Linux because Windows poses too many challenges with finding software, installing software, keeping software working, keeping it up to date, and then the software is just inferior. If you have apps that are specialized (you said Chrome and Thunderbird, I don't know if that's a restriction or if that's just "these are my browser and mail app" statement and you use a bunch of other stuff), you obviously need the OS they run on.
Beyond that, switching to Linux won't keep an aging machine alive. It eats resources too. Put some more RAM in or swap in an SSD as your root drive sure, $50-$100 and it's a big return on investment. You're back to installing Windows over with the SSD (which brings you to the "Maybe Linux" fork, implications above--it's not likely to lower your resource usage significantly), but it's a huge