While headspace should be checked, it's a far overrated concern for those of us who do not reload our cartridges.
Headspace is the amount of extra distance between the casehead (base of the round) and shoulder (front edges of the cartridge, surrounding the bullet). When fired, the cartridge will stretch to fill the available space. The reasoning is that with old, brittle, soft, or poor quality ammuntion, the cartridge case could rupture and spew hot gas out. Also, for some cartridges that are straight-walled, too much headspace means that the cartridge will shove forward in the chamber when struck by the firing pin, but won't ignite, or might even break an extractor.
These things are bad, if they happen.
However, most military rifles are built to generous tolerances, and meant to function with enough slop to endure dirty ammunition and poor maintainence. The 7.62x39 cartridge in particular is meant for those conditions, as it has a tapered case that wedges into place when chambered, and thus minimizes both dirty chamber and loose chamber/headspace concerns.
Thus, if you're not going to re-use cartridges and squish them back to proper shape and fire them again, causing repeated stress on the metal of the case, you don't particularly need to worry about it.
Furthermore, the 7.62x39 is not a round that puts out a tremendous amount of pressure compared to the strength of the casing.
You should be fine but you might want to check anyways. Gauges are cheap insurance. Also, a single layer of electricians tape on the base of an unfired cartridge is sometimes used as a rough guesstimate gauge. It's not pretty, but it does work. If the thickened round won't allow the bolt to close, you should be fine.