Thanks for the continuing tips. I glued a second layer of 1/8" birch over the first one, after sanding it, to get a fresh start. It looks much better. I'll be sure to thin the first coat of new varnish after letting the stain dry well.
Today I learned another possible reason the varnish didn't adhere well. I used a water based varnish over oil based stain that probably didn't dry long enough.
I thought the Varathane stain was water based. Today when I took stained wood samples and a can of stain to a paint store Mike (the store manager) looked at the can, which said to clean up with paint thinner, and told me the stain is oil based. Oooops. I most likely didn't let the stain dry out long enough - wrongly thinking it was water based.