re "which one (module)..." to change:

1. if, as kcott inferred, the change is at your hosting service or by your syssadmin you'll have to take his advice: talk to the supplier. Your observation that you can't reproduce the error on the dev box (I assume that's what you meant) reinforces, for me, anyway, the notion that 'the other guy' made a change, but didn't tell you.
2. if the issue is related to an update (by others) calculated to minimize opportunities for an exploit, and you actually control the production box, you might want to check and see if there's any kind of auto-update process that you've not noticed (logs may help as might other sleuthing thru the software, OS included.

Finishing the thought: OTOH, suppose the change is at bitly -- that would explain their rejection of an older SSL protocol. Again, check the source....

And finally, the Perl Porters go to great lengths to avoid "breaking" things; backward compatibility (at least to 5.8 or maybe 5.6) is a high priority precisely because some major users are so slow (reluctant?) to update. What you're talking about is not quite as ancient, but both your Ubuntu and your Perl date to well before the discovery of the SSL problem (or others which come less readily to mind).