How sturdy is the paper towel? If 2-ply, can you very carefully remove the bottom layer of the paper towel without harming the seedlings that are sitting on the top layer or have they rooted through the paper towel? If you can remove the bottom layer without harming them, you might try that and then just place the paper towel layer with the seedlings on some new soilless potting mix. Then spritz them with a fine mist of water from a spray bottle to help the paper towel make good contact with the potting mix. I'm thinking the paper towel should start dissolving with a few waterings. If the seedlings are far enough apart then you could cut the paper towel up into pieces to help spread them out a bit.
You can always try removing a few of the seedlings from the paper towel to separate pots first. Then when you've transplanted enough of them that you've gone cross-eyed, you could try putting the rest on the potting mix still on the paper towel. If they germinated like mine did, you'll never be able to pot all of them up individually because there will be hundreds. Trying multiple ways should increase your odds of succeeding with some, at least.
They really are tiny seedlings so what I did was just to let them fight it out until they started forming small tubers after a few months and then separated the tubers. You don't save every seedling that way but you will still end up with a lot.
If you succeed in growing them out and collect seeds one day, I've found that folded pieces of white tissue paper work well for holding the seeds. Then I put the tissue paper in a tiny plastic baggie or other small envelope. Don't put seeds directly in the plastic because they will static cling themselves onto it and hold on for dear life!
