Car won't shift into 3rd gear.

Get it scanend by a mechanic who has a al system scan tool - the ones the auto parts stores use do not catch tranny codes in most cases. You really do need codes for this.

That being said... The trans was the weak point in the old pushrod V8 GMs like the Trailblazer - the 4L60 is a decent trans but they are not as forgiving to wear and towing (without using the towing mode) as others are, and it tends to destroy them sooner then say, an Aisin trans. If it struggles to pick up gears, it's not a guaranteed failure but the 4L60 is not the strongest tranny GM ever made - the 4L80 is better and it generally bolts onto 4L60 vehicles, but you need to reprogram the computer to use it. The best ones from this era are the Allison automatics found on the diesels, but you can't exactly drop a Allison in place on the gas ones without adaptation :-(. It also requires completely custom tuning, as the gas GMT800s were never coded to take it.

The motors are bulletproof even past 200k - the trans is the weak point. It could be as simple as the torque converter or valve body but if those are fine it's rebuild time.