HVAC Vents won't switch?

Yep, this is a classic vacuum related problem, I would start under the hood to see if the small vacuum line that goes from the engine to the vacuum storage container to the firewall is in good shape, this guide can help:

https://www.2carpros.com/articles/air-vents-stay-in-the-defrost-position

There is a vacuum rotator valve inside the HVAC control head that falls apart as well. Here is what it looks like. Please upload a short video in your response of the problem. Check out the images (below).

Being a Blazer you have a few things to check. The vacuum line into the canister is one but any vacuum leaks will cause problems, one of the worst is the hardest to get to, that is the line that goes under the battery to the front axle actuator. The fastest way to tell if it is inside or outside is to use a hand vacuum pump and connect it to the line that goes into the vacuum canister, Then apply vacuum, try the controls, does it now switch? If yes it's under the hood. There are a few lines that can be the issue, one to check if you worked on the 4X4 is the triple line that goes onto the transfer case that sends vacuum to the front actuator. One is vacuum, one is the line to the front axle and the other is a vent. If it was plugged on in the wrong location it could cause both the sluggish actuation and the interior problems. FYI, there is an aftermarket kit you can install that replaces the vacuum actuator under the battery with a motor. Makes it switch much faster. Attached are the relevant vac diagrams.

Yep, check the vacuum source at the firewall, this is where the failure occurs.