Checking your draws to the aggressor is standard; leading out into a range that hits the board well is a significant mistake.
Leading into the preflop raiser here is a mistake. We should check our entire range to the CO, who has the range and nut advantage on this Jack-high texture.
Note: Leading out (donk betting) with a draw is generally incorrect; checking allows us to realize equity more efficiently and protects our checking range.
Once we are raised, we have a mandatory call. We have an open-ended straight draw and are getting excellent pot odds to see a turn.
Checking is the only play here. The Ace is a much better card for the CO's range than ours, and we must continue to play defensively.
We must call the small turn bet. With an open-ended straight draw and getting 4:1, we only need ~20% equity, which we have against their range.
Checking the river is standard. Our draw missed, and while the board paired, we have no showdown value and no reason to turn our hand into a bluff here.