Your plan for using builder's string is good, but don't set the string tight. That opens you up to 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" and 1-1/2", respectively). Then you'd measure the gap at intervals.
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).