Here's my suggestion, based on the actual circumstances that earned me the Specialist achievement.
Play defense on pl_badwater. Make sure your team has a few competent damage sponges (Heavies, Soldiers, and Demomen) who know how to retreat (you may need to explain that bullets are hazardous to your health).
You want to stay alive. You can facilitate this by always sticking to a buddy. The best buddies to stick to are the ones that hang around sharp corners near narrow corridors:
In these locations, you and your buddy can stay behind cover in the yellow area, and once they are fully healed (to 150%, conditions permitting), they can venture into the red zone for ideal damage exchange along the highlighted lines of sight (where Heavies, Soldiers, and Demomen have a captive audience, so to speak, of the cart's escorts).
Note that the general advice here (be on a good team, play against a bad team, heal everybody) still applies, and that you'll still need classes like Engineers (who can help by putting dispensers near you) to cover your back.
According to this thread posted by one of the developers on the Steam forums, the achievement for sector 3 is broken and will not unlock, along with
- Finale
- Find the Developers
- Tough Going Part 1
- Tough Going Part 2
- Fizzicks
EDIT: The achievements appear to be fixed. The achievement for Sector 3 can now be obtained simply by playing through the sector like with all of the other achievements. I tested and every one of the previously broken achievements can now be obtained.
Best Answer
Yes, you can get the "Diamonds to you!" achievement in singleplayer. There are 2 ways to get it. The first is precisely what you describe; you can throw a diamond at a zombie, and if they pick it up, it counts for the achievement. Not all zombies will pick things up though, so you may need to try a couple zombies.
The other way you can earn this achievement for a singleplayer world is to open the world to LAN. This may not be possible if you don't have another player, but it is an alternative method if you do have someone else that can connect to your world over LAN.