Agree: the problem is almost certainly that the cable is frayed inside the housing or inside the shifter or the cable has corroded inside the housing or both. Buy new cable and housing and look at some videos for guidance in how to cut the housing to the right length, how to ream the cut end of the housing to ensure that the cable slides freely through, and how to readjust the derailleur once the cable and housing are in place.

On the other topic: cross-chaining doesn't matter, unless it does.

Small-ring-to-small-sprocket usually won't hurt anything (unless the rider uses that combination much of the time, promoting premature wear of the small sprocket, or the chain falls off frequently due to the windup spring in the rear derailleur being too relaxed, or the chain scrapes on the inner side of the outer ring, or backpedaling unships the chain, etc.).

Big-to-big also usually won't hurt anything, unless the chain is too short, in which case the consequences can be dire (broken rear derailleur, bent or broken derailleur hanger, etc.).