I'll sell you mine for $9 each and throw in shipping!
$10 each is ridiculous. A set of 24 is about $30.
And no - you do not need new ones as everyone else has said - unless you want them to be a different color or something.
If you switch to most aftermarket wheels with conical lug nut seats - then yes you should buy new lug nuts. Technically the OEM ones work on both because they also have conical faces on the ends so they work with the spare tires - but they're not the right lug nut for a conical seat wheel. But - again those lugnuts are only around $30 for a full set.
My only suggestion is not to go with some oddball lugnut style that requires a special tool or key to remove. And then go to somewhere remote - and not bring the special tool. I've been there with someone broke down on the trail in BFE without a good way to remove the wheel where the broken axle shaft is. We did get the lugnuts off with a hammer and a socket, but it wasn't ideal. Don't do that.