If you found it, don't lose it now
Solving a puzzle is hard, obviously. But people have solved one and then had it taken from them. #66 and #69 were both stolen by bots at the exact moment the solver tried to move the coins. Once a transaction is in the public mempool the public key is out, and because a puzzle’s range is known, the private key follows within seconds.
So this tool signs, and then only lets you submit through a safe route. If the address is a puzzle whose public key is not yet exposed, the public broadcast button does not appear at all. It exists because guides tell you not to use something like coinb.in without ever offering an alternative. Rehearse the whole thing before you find anything, using the published key of a puzzle that is already solved — doing it for the first time with real money on the line is the most dangerous version of this.
With an ordinary wallet, broadcast publicly as you always would. It is safe with the public key exposed, that is how Bitcoin works, and it is usually over in ten minutes.
The exception is a puzzle. Its range is narrowed, so the instant it hits the public mempool a bot recovers the private key and takes the coins — that is how #66 and #69 were lost. So the public broadcast button is hidden only for puzzles whose public key is not yet exposed, with a private route in its place. The address you enter decides which.
A fresh address from a hardware wallet is best. Avoid exchange addresses.
Raise it if you want it confirmed quickly, lower it if you're not in a hurry. RBF can raise it later.
Look up the balance first, or enter the UTXOs by hand.