How to check if joists are on plane?
Your plan for using builder's string is good, but don't set the string tight. That opens you up to cumulative contact error. You'll end up with a soup bowl for a ceiling.
Instead, put a spacer of a convenient thickness under the string at the outside. I've often used a scrap of 1x or 2x lumber (3/4&quot; and 1-1/2&quot;, respectively). Then you'd measure the gap at intervals, either by setting a series of lines across the room, or by swinging a single line from a corner across to various points along the opposing wall.
If you find high or low spots, here are a couple remedies...

For low spots, sister blocks or strips to the joist as needed, or use shims under your strapping. You can still screw your strapping to the joist, so these don't need to be thick enough for screws. They just act as a spacer.

For high spots, chalk a line from end to end on one side of the joist, anchoring at the bottom corner. Use a circular saw or power planer to remove the hump (sag).


Note that level isn't really what's important here. No one can see a mild slope in a ceiling. Flat is your priority.