I don't see a serious voice coil issue, unless the coil is shorted or the former is melted in such a way that I can't see. What I do see is a separated spider, which is pretty easy to fix. Get a new dust cap with your new surrounds, carefully cut the old one off, set the voice coil back into the gap, use some paper or thin cardstock to shim the coil so it's set right, lift the cone a bit so that you can get to the base where the spider used to attach (if the shims are the right thickness, there will be enough friction to hold the voice coil wherever you set it in the gap), use a long, stiff, thin tool (I've used a fairly thick piece of copper or steel wire, with a small "foot" bent into the end of it) to apply a thin bead of 5 minute epoxy to the spot where the voice coil former attaches to the cone (be careful, you just want enough to fully reglue the spider to the coil, excess can get into the gap and really screw things up), then press the cone down so that the spider re-seats and glues back where it was. It's a lot easier than it sounds. As long as the coil is good, there is no need to replace it, despite what the Negative Nellies may think.
As an aside, this is what can happen when someone continuously cranks up a driver with a bad foam surround...