Kind of a stretch but even counterfeits, especially of an older variety, can occassionally be of some value depending upon their own history...
They are not counterfeits, they are modern copies made on thick, parchment like paper. By modern, I mean since the 1940s, the American Historical Products Company (at least I think that is their name) has been making copies of Confederate currency, Bank of the US notes, Texas notes, etc. on this type of paper. You can find the notes at historic sites. I found a group of their copies of colonial notes at Mount Vernon gift shop.
These remind me of the first time I was duped in this hobby. I bought a note in a frame much like these, (passing up some authentic foreign currency) at a yard sale. When I brought it in I was told it was an artificially aged copy. Artificial aging of these notes is commonly done by, you guessed it, heat/fire. It browns the paper, makes it brittle and appear old. Aside from the ink on the note not matching in color with the example you found, the ink used for the signatures was different, and being lead based, this ink would age to yellow and more rusty colors. Hope that helps! Edit: wow, sorry for the necromancy here I thought it was a new thread. Oh well!
Maybe a more accessible list of modern CSA copies: https://oldcurrencyvalues.com/Fake_Confederate_Money/ ... apparently does not include all the serial numbers of the 71 examples reprinted in 1995 by the "Confederate Treasury Company". Which however, are very clearly marked as Copies on the reverse ("T-21", "T-43". etc)
I do genealogy and collect currency too. I would go by your state archive, and talk to a conservatoire there, and ask many questions as possible as how to conserve them, if possible. let us know what you find out.