Through a couple theories at you.
1. The Axle fluid was so bad that it actually was causing insane amount of resistance on the powertrain which caused your clutch packs to burn/shred/disintegrate "take your pick of wording". ( Super unlikely )
2. Now you stated that 2nd, 3rd and Reverse are the bastard gears after a few miles. Experience is leading me to point a few fingers at your transmission. What I have experienced with is majority older but a few newer 3spd transmissions. So here we go, the sluggish feeling or you feel like your towing the Titanic is from the Transmission fluid, Clutch Packs, and Shift Solenoid or Pump.
If fluid is old it will not have the proper viscosity that the pump and solenoids need to regulate the fluid pressure. This will lead to your jerking and sluggish feeling. Why you ask, I like to call it burping. If the Fluid is old when the pump pulls the fluid it can but NOT ALWAYS give you nice little frothy air bubbles. The bubbles get pumped throughout the transmission once they hit sweet spots like Clutches, Servos, heck even the planetary gear they can but Not ALWAYS do what I like to call burp. This is where the bubbles hit these areas and cause pressure to drop "get it burping, I crack myself up" and catch. This can lead to premature clutch wear and gear damage. Also you could just have low fluid or a gummed up filter.
Not meaning to insult your intelligence just want to broaden your horizon. The information that you gave doesn't add up to a Axle issue that would give the results you have stated.
