The symptoms you're describing do point to the shock absorbers, but they're not always the only culprits. The front end sagging when braking could be down to them, but you should also check the condition of the springs and struts - sometimes they wear out together. Poor handling over bumps - yeah, that's a classic sign of worn shocks. The car kind of "floats" down the road instead of cleanly handling the bumps.
To figure it out more precisely, check a few other things. Park the car and push down on the roof above each wheel - if the body bounces and sways for a while, that's a pretty clear sign. Also pay attention to uneven tire wear - with bad shocks, wear is distributed unevenly and the edges of the tread wear out faster. And check if there's any oil leaking on the shocks themselves, though that's already a pretty obvious sign.
In any case, it's better to take it to a service center for diagnostics and ask a specialist to check the whole suspension, not just the shocks. Maybe you need a complete replacement, or maybe just a wheel alignment will help. Driving with worn shocks is dangerous - your braking distance increases - so don't put it off.