Likely control arm bushings or similar. Get the wheels off the ground, grab a pry bar, and start looking around for damaged bushings and excessive play.

Occam's Razor principle of car troubleshooting, start with the cheapest and easiest before you get into complex stuff like the trans

1) shocks/struts. Yours probably has struts. At 10 years they definitely could be sagging despite the low miles, and you would only really notice at hwy speeds or when cornering hard. OEM quality is $90-140 each. Replacement is a 1-hr DIY job; 5 bolts each, only need to remove the tire

2) ball joints/control arms. Ball joints are cheap, but your lowers will have the awful mushroom fitting method, so it's easier just to replace the whole control arm assembly for $150-220 each. 3-5 hr DIY job per side, requires a lot more disassembly including a few speciality tools like a 36mm axle nut socket. Mechanic will gouge you, but might be worth it if you aren't comfortable or dont have things like a torque wrench. Also will require an alignment after

3) now get into stuff like torque converters. If you have the awful 8-speed, the torque converter is a possibility. The 6-speed is much less likely but possible. Those Chevy transmissions generally start to stutter when accelerating when the trans starts to go, especially around 40-45 mph

ETA: an auto parts store can read your trouble codes for free to see if there is a trans code, you can also monitor the trans temp in your dashboard info center. 180-220 is normal, hotter than 240 is a problem

This is a pretty good new band name.

Trade it in

I'll be damned - replaced the struts this evening and drove it up to about 85, the shakes are completely gone