up to your ears in something
up to your eyeballs in something
up to your eyes in something
up to your neck in somethinginformal
have your hands full
have something on your plate
in the middle of something
be chasing your tail
run round in circlesUKinformal
run round like a headless chickenUK
active
on the go
on the moveinformal
(as) busy as a bee
tied up
unavailable
engagedformal
have (got) something onUK
congested
gridlocked
backed up
bumper to bumper
jammed
snarled upUK