How to get broken screw out of wall anchor

Question

I am putting up a single shelf on a wall with some antique redwood interior paneling, so my goal is to keep the walls as in best condition as possible with minimal additional drilling done (if any).

I put in all 4 anchors for the mounts loosely tightening the screws until I had everything in place. Then I went to fully tighten the screws - due to the layout I'm doing the tightening by hand. So far when I went to make the screws tight two of the heads broken completely off.

Now I have the main body of the screw about 1/8" into the anchor and am unsure the best way to get it out.

I tried drilling it, but I didn't seem to make much progress.

I've looked up alternative methods, but I'm at a lost as to what the best route would be... primarily due to 2 things: (1) how deep the rest of nail is and (2) my hope to keep the anchors there without drilling in all new ones.

Update:

Picture of anchor used (drywall anchors, so not the best ones for the job here).

Answer 1

Those anchors are meant for drywall and use in wood may make them too rigid to accommodate the screws. It is also essential that the right size screws be used.

In general, anything but modest weight pictures or mirrors call for stud mounting, if possible. If not, toggle bolt connections, especially the newer strap style, are recommended.

I would remove the shelf, and back out those anchors. Restart with either screws in studs or toggles.