Cables don't actually stretch. That's a myth. What happens with new cables is that over the first few weeks, they will become more squarely seated in the housing and ferrules due to use (tension), and so then the prior indexing set at installation is then making the cable too slack. So the usual thing is to tighten the cables by turning the barrel adjuster counterclockwise (towards the frame) a bit until shifting returns to normal. In one rare case, the mechanic actually installed my cable too tight, as I discovered when turning the barrel counterclockwise made things worse. If you start running out of room on the barrel adjuster and things are still not right, then the cable is nearing death, or you need to do a big overhaul if it's new. Either way, it helps to loosen the barrel adjuster again (so you have room to re-index). Then grab the cable under the down tube and yank hard sideways to make sure everything is seated properly. Then use a clamp or fourth hand tool to hold the end of the cable, loosen the pinch bolt, take up the slack in the cable, re-tighten the bolt, and then redo the indexing using the barrel adjusters.

If the cables are near death, then you may find they are spontaneously loosening. Re-tightening the cable may only be a temporary fix. Shifting problems will then spread across the cassette. You may also notice fraying of the cables at the ferrules. You should also check to see if the housing has been damaged.

If the issue is just the chain, then skipping only on a specific cog or two suggests that those cogs are unduly worn and the cassette will need to be replaced too.

Because you are experiencing autoshifting as well as skipping, I think there is something going on with your cables, particularly if you're starting to notice hesitation when you try to shift one direction (or both). One explanation is that you've got too much friction going on around the BB. More likely with the age and mileage, I think it's time to replace the cables. That costs like $5, plus new bar tape. Even if that doesn't fix your problem, it isn't going to hurt.