Getting a weird vibration at highway speed after shimming the rear axle - normal break-in or did I mess up the pinion angle?

If it smooths out when you let off the gas, that's pointing at the driveline, not the tires. Tire issues are speed dependent and don't care whether you're on the throttle or coasting. A pinion angle problem is ussually load dependent, so hum and shake under power that fades on decel is the classic sign. I chased almost the exact same thing on my crawler after a spring swap and a new double cardan, and it turned out I had shimmed it too far the wrong direction for that style of shaft. That's the thing about double cardan shafts, the setup is different than a regular two-joint shaft. With a double cardan, most of the angle change gets absorbed up front at the centering yoke, so you actualy want the pinion pointed up a couple degrees relative to the transfer case output, not zeroed out. Dropping a 4 degree shim under the axle without measuring first can easily rotate the pinion too far and put a big working angle on those u-joints. Four degrees is also a lot of change all at once if the pinion was already close. Easy way to check without spending money: download a digital angle finder app on your phone or grab a cheap magnetic inclinometer from any parts store. Measure the angle of the transfer case output yoke, then measure the pinion yoke with the truck sitting at ride height on its tires, not on jack stands, since the axle rotates as the suspension droops. With a double cardan you want the pinion a couple degrees less than the output angle, pointed up toward the transfer case. If you're off by 4 or 5 degrees either way, that's your hum.

Also make sure the slip yoke on the new shaft is fully seated and the u-joint caps are all the same series and fully seated in the yokes. A slightly cocked cap will do this too.

The tires could definitely add some noise, but for now do the coast test on a quiet road at about 50, in gear and out of gear, and see if the shake follows the throttle or just the speed.

Had the same hum show up on my rig after swapping shims around, and it turned out the carrier bearing took the hit. Once the pinion angle changed, that center support was sitting at a new angle and the bearing started whining under load. Worth crawling under there and looking at how the rubber in the mount is sitting now, and whether the bearing got nicked when the shaft came out. Mine was noticeably tilted afterward and a fresh bearing plus a slight tweak to the mount angle quieted it right down. Cheap check before you go chasing shims again.

Still learning my way around here so take this with a grain of salt, but when I put a new rear shaft in my old 4runner the shop forgot to grease the slip splines and it made a low hum under load that had me convinced the angles were wrong. Couple pumps of grease in the slip yoke and it went quiet. Might be worth a check since yours is a new shaft too.

Also curious, does the hum show up in 2wd only or in 4wd as well? Mine only did it in 2wd, wich helped me narrow things down pretty fast.

Another thing to look at is axle wrap. A 4 degree shim rotates the pinion but it also changes how the leafs load up, and if your springs are soft the axle will climb under throttle and rotate that pinion even further off. That matches your symptom exactly, hum and shake under power that goes away when you back off. I had this on my rig after adding 2 degrees, and the fix was just a set of anti-wrap bars off the housing. Easy way to check is have someone hold it at 50 in gear while you watch the pinion from outside, or chalk mark the pinion flange and tire and see if the marks drift apart under load.

One thing I'd check on a brand new shaft is the length. If it's even a half inch long, the slip yoke can bottom out under droop or bind up under compression and it'll shake like crazy right in that speed range. Chalk the yoke with the truck sitting normal, then flex it out on a ramp or ditch and see how much travel you've actually got left. Mine came up short on travel and it hummed til I had it shortened.

Also curious, does it do it in 4wd too, or only in 2wd? That'd tell you if the problem's coming from the rear axle or somewhere upstream.

Update: turned out to be the carrier bearing. Crawled under there after reading the replies and the rubber in the center support looked cracked and squished from sitting at the new angle once I shimmed it. Swapped in a fresh bearing and the hum is gone, no more shake at 50 either. Rides smooth on and off the throttle now.