If you're going to look to take it apart, I would try to remove the core from the nut before buying a new core. There was no way i was going to get the core out of mine - and many have similar stories of trying to get them loose in the nut. Short of taking high and directed heat to it (others have without success) the core can be a real PIA to get loose. I ended up taking the whole assembly out and sanding it down - having to fit it into the pf periodically since I couldn't get it loose to adjust the position of the core.
If you can get yours out, buy a new one, fit it proper in the pf and be done with it. If you have to sand it, take your time and reduce grit levels as you get closer to level so as to get a polish to the top. It should be level with the pf to avoid wear to both.