How do I safely remove a drywall anchor from a stud?
If the anchors are wedged tight in the wood, full length, you don't really need to remove both of them, you can use leave one in. Remove just one to make everything level.

To remove it, just use a wood drill bit slightly larger then the hole and drill the plastic anchor out. Do NOT drill any deeper than the anchor length.

You can now glue in a dowel, pre-drill a pilot hole and screw in a screw at a correct height. Use a longer screw, say 3 1/2" or so, so at least some of the screw is inside the undisturbed part of your stud.

Personally, I'd just leave the anchor where it is, drill a new pilot hole where it needs to be (right through the anchor body), and use a longer screw, and not bother with gluing dowels. I wouldn't trust these glued-in dowels anyway. If your new hole needs to be slightly lower, I suggest you go with this route.

Another solution would be to abandon both anchors altogether (cut their bases flush so you can cover their holes later), and drill two new holes in a slightly different location (if this is possible in your circumstances).