Headset fully tightened, but fork still loose and wobbly
The wobbling could come from the fork. This is easy to check by putting a finger between two parts of the headset (as you can see here, around 0:35 ). If you brake and feel movement, problem is for sure on the headset.

If you do, is the headset the correct size, BOTH for the frame and for the tube? And are they in good shape? These parts have low tolerances because they have to fit the frame and the fork tube very tightly, and different manufacturers have slight different dimensions.

If you can, go to your usual shop and ask them to test with another one headset.

Good luck. And until you fix it, please don't ride that bike.