I’m a bit tied up right now.
I’ve got a lot on my plate today.
I’m slammed. / I’m swamped.
I’m stretched too thin.
I’m up to my ears/eyeballs in work.
Let me get back to you in a bit.
My schedule is pretty packed right now.
I’m working on something urgent at the moment.
I need a little time to finish what I’m doing.
I’m juggling a lot right now.
I’m overwhelmed / overloaded.
I've a lot of irons in the fire right now.
I’m at maximum capacity at work.
I'm at full bandwidth right now.
I’m buried.