Being in a rush really doesn’t have any connection with swamped. You can be in a rush without being swamped. If you’re in a rush, it simply means you’re short of time.
Sorry, I’d like to chat longer, but I’m in a rush. I’ve got an dentist appointment in 10 minutes, so I have to leave now.
Swamped means that someone has an incredibly large amount of work to do. (Or has deal with an incredible amount of something.) So much that it might seem almost impossible to finish or deal with it all. The person might be overwhelmed by it.
He is swamped with work. He’ll probably have to work overtime for the next 6 months to get it all finished.
He was swamped with requests for help.
Be up to your eyeballs in (something) is an idiom that means 'have a huge amount of ’ (usually work). This expression has no literal connection at all with an ability to see.
I’ve also heard this idiom as be up to your ears in (something). It has exactly the same meaning.