Yes, that's normal. Sleeping doesn't immediately get the alcohol out of your body or cause your body to process it magically.
You can also be dehydrated and other feelings contributing to drunkenness.
When I drink I usually get up early the next day, it's just a weird habit of mine especially when on vacation so there have been times where I've been pretty shitty at night, and gotten up like 4 hours later and still definitely felt it. for me I usually just pound a lot of water, eat a small breakfast, take 2 ibuprofen, and usually exercise, and once I shower and stuff I feel better.