You can’t dissolve the wire without the coin dissolving as well. You
might want to take them along to a coin dealer first to make sure
that the final price for the coins will be what you expect. People who
collect usually want coins in really good condition and the price
drops rapidly if there are signs of wear and usage (circulation).
Simply wearing them or having them rub up against other items in a
jewelery box will have marked the coins and they will no longer be in
mint or even very fine condition. Most settings do mark the coins,
I’d suggest that you take it to a really competent jeweler and get
them to cut away the setting/frame. Then you can make decisions about
selling them or keeping them for sentimental value.
I don’t like giving you a downer and I hope they are in fine
condition but my experience tells me that they are probably not going
to fetch what you want.