I put wall anchors & screws into my wall to hang a picture. The picture is not level. How can I level it without removing the anchors (will casue large holes) ? I tried to bend the screws up or down at an angle to change how far the picture sits.

Screw D rings to the back of the frame a third of the way down from the top, both sides. Then use picture framing wire to attach to either D ring. D rings are rated for weight as well as the wire and a hardware store or frame shop will have them. This is the professional way to hang a picture, and it will not matter if the wall anchors are level, the picture can be adjusted right or left to find level.

The only problem is that the frame is a clear 16"x19" with a 12"x13" picture centered in it. I'm worried you'll see the wire.

Ok, I understand. What type of fasteners is it going to hang from? Is there a way to either move one of them slightly or use some tape or other wire to bulk up the hardware to compensate for the slightly different/uneven wall mount?

Because of the drywall (not on studs), I put in the wall anchors. I put screws in the anchors & tried to hang the picture off them. The frame has the standard "teeth (vvvvvv)" hangers, (customary with picture hanging) already attached.

The sawtooth hangers are either nailed on with little tiny brads or are bent metal, and the sharp end points penetrate the frame. I think the easiest fix might be to raise the low one, or lower the high one just a smidge to adjust.
A level across the wall anchors will show you about how much adjustment is required.

The picture sounds like it might be heavy with glass, seems unusual to use the sawtooth anchors for hanging if it is heavy. These are often used for lightweight wall pieces.

Keep in mind that many wall hangings that just hang (you don't hang other things on them, like shelves or coat/had racks) probably do not need wall anchors and can often just hang from one inch wire brad nails. I put a nail in the drywall at a slight downward angle, and when removed or adjusted, only leaves a tiny hold that is hardly visible or easy to fill.