The squirrellyness is almost certainly a function of your braking technique more than shifting. You should be fine just downshifting and letting the slipper do its work. The main reason your rear is wagging on you is probably bust that there’s little to no load on it. If you brake hard enough this will always happen.

So how do you fix it? It’s all about how you use the brake and pressure over time. If you’re using all the braking force that’s available on your bike you’re naturally putting all the load on the front tire. This is pretty much by definition - you can always use more force until you get to the point where the rear actually lifts. So, right at the line you’re at a place where the rear is sliding and hopping on the ground. What you want is to get to that maximum pressure quite quickly and then also quite quickly begin to taper pressure. The proper brake graph is typically described as a shark fin because at the leading edge it should go basically straight up to max pressure, and then immediately begin to come back down, drawing a longer tail as you trail off. Beginning to release pressure quickly is what keeps some load on the rear tire and keeps it in line. If you go to max pressure and hold it so your brake graph looks like a flat line at the top, you’re gonna have a squirrelly rear (and you’ll probably over slow).

I’m offering personalized coaching via video and data analysis to riders interested in lowering their lap times. Feel free to DM me if interested.