If your Mint 4 installation has a separate /home partition you can upgrade to Mint 6 by burning the Mint 6 downloadable ISO (Final (gnome), KDE, XFCE or Fluxbox -- your choice) and keep all your personal configuration files intact. If M4 is all in one partition you'll have to search these forums to find out how to back up your /home files.
BUT, the problem may be a corrupt T'bird configuration file, in which case upgrading, but keeping your personal settings would not help: the problem would still exist! So in the end it ma be wiser to do a fresh install: everything will be new and work perfectly.
It was for this reason I suggested installing kmail and set it up to see if it works: if it does, the problem is not M4, but T'bird itself. If kmail does the same thing, it's M4's problem. I'll point out that kmail is a KDE app; I think the gnome equivalent is evolution, but I'm not 100% positive on that. You could search for "gnome email" and synaptic would tell you exactly. But you could install kmail regardless and in truth, I prefer kmail to T'bird (just my preference). They look and behave and setup pretty much the same. Do that first: it's the fastest way to tell you where your problem lies, and what you should do to fix it.
You could also, from within Thunderbird, set up a new profile and add one email account and see what happens. If that fixes it, you may not have to upgrade at all! To start Thunderbird profile manager, in a terminal type "thunderbird -p" without the quotes. If you want t always start with the profile manager, edit your menu so the command to launch T'bird has the "-p" switch. The same procedure will work for any Mozilla product.
I can give instructions on saving your Firefox bookmarks and passwords too, if you need them due to a fresh install.
If you need specific guidance anywhere, just ask.