My gut feeling is that there is more to the problem than a worn spring or a worn or loose handle screw? I think the real problem is the brass piece that holds the handle is worn where it goes through the body of the lock. In the lock in the picture you can set the end of the handle through the brass piece and with the screw tight you can see movement in the brass piece which would cause a sag. This movement has nothing to do with an old spring. You will notice cracks in the brass in the second picture. The cracks are from me heating the brass and then trying to compress it with a small socket and a vise. No luck, all I did was crack the brass.
I don't know how to fix this problem except for replacing the lock. Like you said, there are a lot of sagging door handles out there, hopefully someone else has a fix? Good luck, Mike
P.S. I did an Advanced Search on this topic with little good advice. You could repost the question to the 37-42 forum, they have a lot more viewers.