There's not a whole lot you can do with a cracked nut. It could shift and cause the problems you mention. You should be able to observe if this is the case. You could possibly try hot glue on it, but if the crack is in a string slot, I doubt you'll get it to hold long.

This if course involves removing the nut in the first place - carefully!. then cleaning the old glue out of the slot, fixing the nut and then regluing it into place. And then it might not hold.

This happened to me on my old acoustic a long time ago, and I replaced the nut with one the music store happened to have. Wasn't even the correct size exactly, but it holds true to this day. You should probably just plan on buying a new nut.

If you are not up to the installation and possible slot adjustment, you can take it to a guitar repair and let them do it.

Try posting in the Guitar Tech forum in the full access section. There might be some good advice forthcoming. :)