How do I get a drain auger unstuck when it won't budge?

If the auger head is frozen, it usually means it's hooked hard onto something dense, like a clump of grease and hair, or it's caught right at an elbow. Don't just keep cranking back and forth like a drill. First, retract as much cable as you can until the head is just past the drain opening. The goal is to give you slack and control over the head itself. Then, try to twist the head back and forth manually, not just cranking the machine. You're trying to break the suction or loosen the grip, not push through it. If it resists, feed the cable back out an inch or two, then try to gently push it forward and spin it, rather than driving it straight in. Sometimes the obstruction is just a solid ball of gunk that needs to be rotated around, not punched through. If that doesn't release it, you can try introducing some water or a strong solvent down the drain. Pouring a little hot water mixed with dish soap or degreaser directly down the line can lubricate the clog and help the auger slip free. Don't just run the machine with the lubricant; let the liquid sit for a few minutes after feeding it down. That gives the soap time to penetrate the blockage material.

One thing to keep in mind about PVC from the nineties is that it's generaly sturdy, but if the auger is lodged right at a sharp bend, you might have just wedged it tight. In that case, you need to relax the tension. Pulling with steady, moderate pressure while rotating backward is risky because it can cause the cable to snap or shear off inside the line. If you feel solid tension, stop pulling and focus entirely on loosening and rotating the head instead. If all else fails and you're short on time, stop, retrieve the machine, and plan on calling a pro for the follow up. They have specialized tools designed for these kinds of stuck situations.

Been there more times than I care to admit. First thing I'd ask is wether you pulled the trap before you even started. Back when I was learning, the old plumber I worked with wouldn't let a greenhand near a drain untill he'd popped the trap off and looked. Half the time the clog is sitting right there and you never need the auger at all.

If you're working through the cleanout instead of the drain opening, you've got a straighter shot and way less chance of the head snagging on a fitting. Also, a garden hose snugged into the line with a rag around it and someone cracking the tap can backflush a soft clog while you ease the cable out. Slow hands win this one, not muscle.

Mine did the same thing in a bathroom sink a few years back, turned out the head had poked through a wad of gunk and the gunk closed back around it like a fist. What got it loose was a spray can of silicone lubricant I had under the sink, shot a bunch down the opening, let it sit maybe ten minutes, then worked the cable a quarter turn at a time. It let go with a little pop and slid right out. Dish soap never did much for me but that silicone stuff is slicker and stays put longer. Also try pulling while a helper runs the faucet hot, the warm water softens grease fast.

Update: the silicone spray trick worked, let it sit about ten minutes like you said and the head finally let go. Cable came out fine. But the sink still drains slow, so I guess the clog is still in there. The reply about popping the trap off first has me wondering, should I do that before running the auger back thru, or just go at it again from the drain opening?