It is easy to check that the adjuster is working. Just loosen the lock bolt: with the engine apart as in that image the mechanism should snap to "fully tight". If you slot the rear blade in place you should be able to push the adjuster back down to "fully loose" with it... lock it there so you can reinstall the cam chain. If it doesn't move freely I would consider replacing it, but the part is not readily available.
The springs: what does it matter? Usually they're weak but pressing on the adjuster rod to get the right tension isn't that tough. And, I don't know that they are available anywhere.