Almost forgot. you can get regular molly bolts that require drilling a hole.
Since you already made holes in the wall. 
These will work great because they are bigger than plastic anchors, You have to drill a larger hole for molly bolts.
Drill out your smaller ones and you’re good.
Another vote for the first kind. Never had a problem with them, solid as a rock and easy to install.
These are my favorite.
I have thin walls (3/8"), so most others, including those neat corkscrew one-piece jobbies, don’t work so well.
I just use a Phillips screwdriver to bore a hole slightly smaller than the anchor and then wiggle it around if needed to allow the anchor to fit snugly.
Then, before tightening the screw (to spread the wings), give the thing a whack with a hammer, to seat the two small prongs on the back side of the flat washer part. That’s what causes it to grip the wall enough for the screw to be turned.
If you have a bunch of these, use a power screwdriver.
I have found that the problem with pounding them in without a hole is that a chunk of wallboard on the back side might rip out, especially in thin and older wallboard.
And should the day come that you no longer want these beautiful anchors, remove the screw and then use a slightly larger drill bit to drill enough for the flat washer part to come off. Then knock the rest of the anchor inside the wall and patch the hole.
Another thought for the OP: Drywall is *real *soft. If you’re not an expert with a drill motor, or if you’re working at an odd angle, it’s easy to wobble the bit as you go in. Which has the effect of making the hiole both bigger in diameter and also non-circular. Either one of which is certain death for the crappy anchors you were using.
So in addition to using better anchors …
Be darn sure you’ve got the right size bit. One size too small is much better than one size too large. Make darn sure you’re going in perpendicular to the surface (both left/right AND up/down) and ensure the motor & bit do not wobble even a smidgen as you drill in. It might take 30 seconds to drill the hole right vesus 10 to drill the hole wrong. That 20 seconds is cheap insurance.