1. Tied up
I'm a little tied up with this new project.
2. Occupied
She's a bit occupied today dealing with new staff.
3. Overstretched
He's slightly overstretched at the moment.
4. Over-extended
She's quite over-extended in this particular role.
5. Overloaded
She's rather overloaded with all these new cases.
6. Swamped
They've been completely swamped with new customers this week.
7. Snowed under
She's been snowed under with complaints.