Actually, one of the flasks is proving to be quite difficult.
I picked the (possibly original) cork out of the flask lid and it consisted of layers of cork disks that were at this point quite dessicated and rotted. The new cork pieces were not deep enough to fit the flask lid, so I carefully measured and cut down a wine cork to fit. It popped into the lid quite easily, but my measurements must have been off by a millimeter or two as I was not able to close the flask lid. Getting it back out again was a struggle, I had to pick it apart with a pair of needle nose pliers, all while holding my breath that I wouldn’t damage the flask in the process.
I’m going back to the drawing board to try to find a more suitable piece of cork or cut another wine cork down a little bit differently.