Carpentry and Woodworking - Resstaining and Polyurethene

First I am not good at any household project, ask me about horses, barrel racing and Boston Terrier's and I can help.

I decided to clean my kitchen cabinets, re stain them, and put polyurethane on them.
I did pretty good with everything but the poly.

When I first put it on the frame around the cabinet doors some type of white stuff showed up as it dried.
I was able to get it off and was not going to put the poly on again, but I couldn't resist.
No white stuff showed up around the frame so I decided to put it on the doors, which looked ok without it, but were kind of sticky feeling.

I was doing all of this in the house since I've heard weather messes up things.

I used a paint brush to put it on one door, but I like old socks better so I used one on the second door.
As I was doing the second door I noticed the white stuff on the first door. I finished the second door and as it started drying the white stuff started appearing.

What am I doing wrong? The guy that lives next door built the cabinets when his brother lived here and used the same stain (color) I am using. I've been letting the doors dry at least 3 days, so now what?

Thanks for any ideas

Welcome to the forums! Did you remove the underlying polyurethane before doing all this? Likewise, I'm a nail driver, not a painter, but we have pros that will chime in shortly to give sound advice. It does sound as if the underlying polyurethane or finish is being melted by the new application since it still remains. Applying stain over the original finish may not give the results you desire, either.

Did you use oil base poly or water based?

How did you clean up the cabinets? Did you strip them down to raw wood?