Draising sounds great, I didn't know it had a name! Do that.

No to the wheels, unless you're on a trike you need to lean that thing over to make the turn anyhow.

Use your initial death grip to hold the brakes about half on so you can use a very high pedaling effort (which will make you feel better) while keeping your speed low (so you can stop when you jam upright and stick your feet down in a panic).

Use a parking lot or cul-de-sac and practice figure-8s in both directions until you feel comfortable. Reduce the size of the 8 as you can confidence. Slower is harder, once you're nice and slow, then let off the brakes and try to relax and go slowly.

After that, it's going to take about 100 miles before you really relax and ride without stress. Start taking some easy 10-15 mile rides.