I found myself in a similar situation not long after I started brewing. For my stepfather's birthday, I decided to make four different beers and bottle them into Mason pint jars with priming sugar (it was a lot of beer and jars just seemed so much easier to deal with). I was aware of the potential issues with Mason jar lids, but decided to try with the plastic screw lids, thinking they might seal in fine. I should mention that this was going to be a BIG beer themed birthday party. We had beer chips, beer cheese, beer movies, beer pong, beer bandoliers . . . so we needed good beer. The party date rolls around and the night before, I'm cracking open the beers and not one had any carbonation. They did prime, but all the CO2 escaped the jar lids (you could occasionally hear a light hiss if you shook one up, and there was no backsweetening). We panicked bigtime.
What saved my butt was buying a Soda Stream. It worked fantastic on the fly and the beer was saved. Bonus was, my stepfather had been wanting one, so he got to keep it as a birthday present. He's still using it to carbonate the tons of leftover beer months later and actually prefers the carbonation from the Soda Stream because he can tailor it to his tastes with each beer. I know there's a lot of stuff on the forum and the internet in general, mostly con on the Soda Stream option (not real carbonation, can make a mess, etc.), but I've been impressed with carbing beer this way and considered getting one myself before getting into kegging.
There's a little bit of trial and error involved: ideally, you would carbonate this way a little bit before you want to drink your beer and just let the bottle sit attached to the Soda Stream body so that the CO2 can incorporate better into the beer rather than letting some escape if you pull the bottle right off. As far as preventing messes, you just take the bottle off slowly to kind of burp the bottle. And it's going to take some practice before you hit on the right carbonation formula for you (how many pumps of CO2, how long to let the bottle sit or not, etc.). But my parents still occasionally bring me beer carbed this way sometimes days after doing it, having let it sit in the fridge, and I'm really pleased with the carbonation. Might be hard to justify buying a Soda Stream unless you like carbing this way, but I know some homebrewers do justify it as a way to sample beer at priming before carbonation. An option, anyway.