I've been here before. Can you double check to make sure you have silt? Draw off the sampling faucet into a clear glass. Does the black stuff look oily and does it smear on the side of the glass? Does it stink? If so, then you have iron bacteria. Sounds nasty, but it really isn't. Actually poses no health risks and given some time the water will clear up. Throwing a couple pool chlorine tables down the casing will help. Just don't put too much down or you'll kill the softener. Honestly, there is no permanent fix but an iron filter will help deliver good household water. I attached some pics of our well water when we first bought the house. It's been 4 years now and the water is much clearer.
Now if the water in the glass settles out to a grit, then you have a silt problem. Aagain, there is no permanent fix as the problem is between either the gravel pack or the well screen. Unfortunately, neither is repairable; the solution is to drill a new well. As a work around, you can add a whole house sediment filter. I actually run two 20" blue filters in parallel because we pump a ton of silt. That said, I've never had to change the filter and it's going on two years now. You'll want to install the filter before the softeners to protect them.
I know what you mean about having to replace a good productive well, but that's what you're looking at for a permanent fix. That said, I've been able to filter and condition my water to a point where the end product is actually crystal clear and quite tasty. Way better option than punching a new hole in.