Wanted to give a final resolution. I got it running normally. I'm guessing that when I put on the new carb, the gas tank wasn't as well cleaned out as I thought it was, and a piece of rust or debris got into the new carb. As well as I found the cap on the fuel pump spring doesn't quite fit the bore that well and can get stuck against the sides (a common problem, based on the Amazon reviews).
So I took the new carb, with the old fuel pump spring cap, cleaned out the jets again with compressed air, and put it back together. Fired up and runs like a champ. I knew it had to be something simple, but was a little perplexed since the new carb was acting up the same way as the old one.
Thanks for the help.