On Ctrl+C, kill the current command but continue executing the script

The continue keyword doesn't mean whatever you think it means. It means continue to the next iteration of a loop. It makes no sense outside of a loop.

I think you're looking for

trap ' ' INT

Since you don't want to do anything upon reception of the signal (beyond killing the foreground job), put no code in the trap. You need a non-empty string, because the empty string has a special meaning: it causes the signal to be ignored.