Here's the process that works every time:

Fill both cups with grease and the correct size and number of balls. Older Schwinns may use 3/16 on the lower race and 5/32 on the top. Fill up each cup and remove one ball. Smooth the grease so none is projecting above the cup.

It's easiest to assemble with the frame upside down, so that gravity holds the fork in place. Either way it's best to have the head tube vertical rather than tilted. Carefully insert the fork column through the head tube until it's seated. Turn the fork back and forth to check for smoothness and to distribute the grease. If working right side up hold the fork in position.

Thread the top race on and turn it down until it is almost touching the balls and grease that you placed in the top cup. Then pull the fork away from the bottom cup until the top race is seated. Now continue to turn in the top race until the assembly is snug.

The above method prevent the race from catching the grease and dragging the balls out of place before the bearing is snug.

Now just rotate to make sure things are smooth, wipe off excess grease and adjust.