Any machine can be used for free-motion quilting. Lower end machines will give you grief with a small throat (difficult to get the quilt through) and mechanics (sturdiness and stitch quality). Maybe you could try doing some smaller quilts on the machine you have. A lap quilt should not be a problem. If you cannot drop the feed dogs, then you can cover them.
Looking for a new machine, look for used machines that were trade-ins. You may get a good deal that way. Make sure the machine has the needle down option. That is an absolute MUST for me. I also like motor speed selection, good lighting (but that can be supplemented with exterior LED light), a large throat, and a good-sized extension table. My machine also HAS to do a good blanket stitch and a consistent straight stitch.
Good luck.