Why is my water softener squealing and shaking?

We just bought a house that has a Rainsoft Silver. About a week after we moved in, the short run of pipe between the water meter and the water softener started squealing and shaking badly whenever there is a big draw of water. It doesn't do it when you just run a faucet, but flushing a toilet or running a shower seems to trigger the symptom.

I figured out that if you close down partially on the main water cutoff, the shaking and squealing stops. Then you can open the main water cutoff all of the way back open and finish the shower or toilet refill without any noise or shaking at all.

I'm also pretty sure that it's related to the rain softener because if you turn the bypass valve to bypass the shaking and squealing stops.

Two other clues:
1) It almost sounds like the bypass valve is sucking or blowing air when there is shaking in that area. The reason I say that it that it sounds almost like a whistle in this area.
2) Not sure if this is related or not, but the brine tank got really low or ran out of salt right around this time. It was really dirty with some silt build up, so I took it outside and cleaned it out well with a hose and scrub brush. I refilled it with salt and have manually regenerated twice in the last 24 hours. The regeneration process seems to be working fine.

Does anyone have any ideas on what may be causing this? Thank you in advance for your advice.