I decided to install center mount drawer slides to fix this problem. Follow along, I'll show you how!
First, pick a drawer that you don't use as much. We want to get our technique down before we get to the problem drawers.
Empty it out. Such pretty liner paper!
Leave these tacks lying face up on the floor. You're going to want to roll your leg or step on them as much as possible. If your home improvement project doesn't end with a trip to the ER for a tetanus shot, YOU'RE DOING IT WRONG.
Separate the two parts of the hardware. They have names but I forget what they are. That's probably the lockjaw setting in.
Realize that the groove that's cut in the back of your drawer isn't wide enough to accommodate your hardware. Think about saying "Eff this," but then remember you have a dull hacksaw! Inelegantly chop a wider clearance in the back of the drawer.
Install your hardware on the back of the drawer. Easy peasy. This would be a good time to poke yourself with those tacks again. Or you can just cut yourself on the screws that now poke through the inside bottom of the drawer, because the wood is too thin.
Stop taking photos at this point because it's so fricking difficult to put screws in straight in such a tiny cramped space. When you're installing the base part of the hardware inside the cabinet, be sure to strip the screw. That way, if it's in the wrong spot, it will NEVER come out.
Marvel at the fact that the one drawer you had that worked well now makes a horrible clicking noise as you try to wrench it open. Decide that it might just be easier to put those little plastic guards in the rut the drawers are forming in the cabinet face.
Plan to make friendly with a carpenter or a cabinet maker and hope that they can give you a good deal on cabinet refacing down the road. Voila!