Will using concrete anchors in wall studs cause problems?
Yes I'm afraid concrete anchors are not appropriate for wood studs. Sleeve anchors rely on being able to press against the sides of their hole with enough pressure to counter act the pull-out weight of the load they're bearing. If the sides of the hole are somewhat squishy (like soft pine is) the anchor will probably fail. You can:


Move the unit slightly and re-drill properly sized holes, just make sure to move them at least 2" away from the old holes.
Fill the holes with 3/8" dowels and glue and then re-drill.
Bore out the holes in your hardware and use a 3/8" lag (or whatever size works with the holes you drilled) if your hardware will allow it.