I have a winter road bike with 10sp Shimano Hollowtech bottom bracket. I used it throughout the winter and retired it in mid February and switched to a bike with bigger tires to handle all the salt and gravel. It is now warm enough to take down my road bike and my crank is literally stuck to the bottom bracket, and there is a lot of resistance as I hold onto the pedals to spin the crank. What exactly is the problem here? Do I need to replace the bottom bracket?

Take it apart, clean it, and see how it goes. It's easy enough to do. I can't imagine the bears would have seized that much for you to have trouble spinning the crank.