Is there a verb form of "busywork"?

Solution 1:

Idling. As a verb:

to spend time doing nothing or nothing useful
of an engine or vehicle : to run without being connected for doing useful work

As Jim said in the comments, "looking busy" is also a good grammatical fit for the example sentence. "Standing by" might work depending on the job, especially if you could be called into action on short notice if you're there. ("Looking into it" or "training" or "taking care of other tasks" might be rated a better answer if this were posted on Workplace.SE).

Solution 2:

I'm gonna say there is not a single-word verb form of busywork. I have candidates such as occupying oneself, keeping busy, and the like, which have connotations of nonsignificance -- but I can't really come up with a single word.

In your examples I would simply use find something [else] to do instead of keep {blank}ing.

It's an hour before quitting time, but I've finished all my work. Can I go now, or do I need to find something else to do?

My boss asked me to look into this, but I've investigated, and there's definitely no problem. Should I just find something else to do?

Solution 3:

putter, or to putter around

(Oxford)
v. 1. occupy oneself in a desultory but pleasant manner, doing a number of small tasks or not concentrating on anything particular:
"early morning is the best time of the day to putter around in the garden"
2. move or go in a casual, unhurried way:
"the duck putters on the surface of the pond"