If it were my bow, I would consider all of those things. The string is a good start with no down side, and will help a bit. The other 2, tempering and piking (shortening) the bow bring with them some risks. The primary risk is totally ruining the bow. I would accept that as I have a room full of bows. Trashing an under performer in hopes of improving it....I have done. I don't have any experience with Hazel but I know others have heat tempered it, and my guess is, it would firm it up to some degree. You again run the risk of doing something wrong since you have no experience, but it's pretty simple and you can be walked thru it here. Piking the bow, ditto. 77 inches is long, really long, so losing a few inches should be doable as long as your draw length is not 32 inches. Again, folks can walk you thru it, just be prepared for the result to be less than you have now. If that is acceptable, get 'er done.
I just read Pat B's answer, and just to clarify, we are in agreement. The string wont make a big difference, but it wont hurt anything either.