a lack of sleep can also trigger depressions. If you really can't take a few weeks off, then you should at least try to get a minimum of 8 hours sleep every night.
And lots of chocolate - it helps you relax.
Still, the vacation thing must be sorted out. You have a right to have a few weeks off and you being the manager means that you tell your underlings what they are to do. No whining tolerated. Be bossier! You deserve it!
And talk with the area manager. He might help you putting pressure on your co-workers (since atm you don't have the nerves for it). It's better you take 2 weeks off now than being ill for 2 months.
And you should by all means see a doctor. Depression is nothing you should take easy.