LF, welcome to UZI Talk. Congrats on the new UZI. Don't give up on it yet!
For the gun to shoot bursts, the bolt has to come back far enough to eject the empty and strip off a new round from the mag but not far enough for the carrier to catch on the sear and thus stop it from firing a second round. There can be a number of causes. As davidl mentioned, weak ammo is a common cause. Try it again with hot ammo. Remember that off the shelf 9mm ammo is generally loaded for pistols and can be quite weak in an UZI.
If that doesn't fix it, then look for something that is causing friction so the bolt doesn't move freely. If the bolt is hindered in some way you'll see the results you mentioned. Things that could cause friction:
- A very dirty gun. Disassemble and look at the bolt and inside of receiver for excessive crud. Clean if necessary.
- Magazine might be rubbing against the bottom of the bolt. If it does it will slow the bolt down. Try a different magazine if you have access to one.
- Bolt might be rubbing on the receiver or top cover in some strange way. To test, remove the magazine and the entire lower (by pulling out the one push pin that holds it in place. Take the bolt out of the gun and remove the recoil springs. Put the bolt back in the gun and title the gun forward and backward. The bolt should slide freely. Put the top cover back on and do the same thing. You should be able to hear it slide freely.
If it's not one of these, the problem could be slightly more serious. It may be that the bolt is coming back far enough but the sear is not holding back the carrier. Again that could have several causes:
- Could be wear of manufacturing problem with the front edge of the carrier.
- Could be wear of manufacturing problem with the sear.
- Could be that the lower is not positioned correctly. This has happened before. If the front end of the grip assemble hangs too low away from the receiver, the sear might not be high enough to catch striker.
Give these things a try and let us know the results.