Doherty Plumbing
Journeyman & Gas Fitter
This is the best I can do.
Yes replace that limit control you took a close up of in the 2nd picture.
All you have to do is basically wire the new one the exact same as that old one.
On your old limit you'll notice that on the right side of it (when looking at it in the orientation it's in now) you'll see a brass jumper. It will either be intact or broken. Make sure you the new limit matches this. IE if it's broken break the one on your new one, and if it's not broken leave your new one alone.
This jumper is used to determine if you have 120v or 24v controlling the circuit.