The last sentence is wrong as written. 더러워서 씻어요 is correct, but 더러워서 씻어서요 doesn't make sense. Since -서 indicates a reason, it is like "Because I wash my hands because they're dirty", and we don't say it like that.
The second sentence is a little strange too, although not completely wrong.
1. 손을 씻어서 욕실에 있어요. (strange because -서 indicates a state that is a reason for something)
2. 손을 씻느라고 욕실에 있어요. (natural. -느라고 shows an action that affects another).
3. 손을 씻으려고 욕실에 있어요. (natural. -려고 indicates a purpose).
They roughly correspond to (1) I'm in the bathroom because I have washed my hand, (2) (because I'm) washing my hands, (3) to wash my hands.