Weird gyro effect while riding
The side to side you're feeling is that your headset is binding up. This can either be because it's too tight or because you've put some part of the headset in upside down (i.e. a bearing cage or installed the cups the wrong way around) or something is out of alignment. It is especially a problem on the bottom cup, when the headset is under load and can't micro-adjust the wheel as you ride.

Check everything is seated properly and in alignment (try putting a sheet of paper between the cups/races and the frame. Check all your bearings are clean and lubricated and nothing is installed backwards.

When tightening your headset you want it tight enough that there is no play, and no tighter. To check for play put the bike on the ground, hold the front brake and push/pull on the handlebars. Then do it with the wheel turned 90° left and right. Ensure you're not tightening the top race when tightening the lock nut.