Can't help you with the sauna, hammam or shave, but for a boat trip or day trip to Greece, see Ahmet at the BBC (Bitez Boat Cruises). He's a couple of stands along the front, one outside the mosque and one down by the Manuela Hotel.

We're just back from Bitez. Hammams are ranging from £4 each for the basic to around £30 for the full works with a massage, face pack etc. The offers should all be checked out as the barber will try to sell you the works and get more profit.
Going to Kos? This is about £20 each, but we were annoyed by Kos Tours who run all the transport frrom the hotels which are arranged by the Travel shops. They had all the passengers waiting for their service on return wait in a group until all were off the ferry, then parade us the full length of the marina, along the front and up a side street to a private car park, about a mile altogether, where the buses were waiting for us. One passenger, walking with a stick, really struggled to go the distance but the guide didn't bother about her dropping behind the group.
I played hell with the TA and informed our rep, who we didn't book with, just to warn him, that this is purely a money saving scheme for the Kos tours firm.
The buses have to pay a fee every time they go on to the Marina, so this was purely to save that fee and pocket extra cash from the victims, sorry, passengers.
It's a great day out in Kos, but DIY is the way to go.
The ferry leaves at 0930, so you need to be in Bodrum by 0815 to walk from the Bus Depot to the Marina, get tickets and a stamp on your passport.

Parky, a very annoyed Parky.

Hi Parky,

Can you give me some more info about Hammams in Bitez, where are they, apart from the Ambrosia and some more details on prices?

Cheers!

Sorry mate, no. There was one in our Hotel, The Grand Newport, and we did see quite a few around Gumbet and Bodrum, just a Dolmus ride away. A couple we met used one in Gumbet and others one in Bodrum, but both came back with the same stories.
All the Travel shops have them in their brochures which you will be able to pick up in your hotel.

Parky

Just back from Bitez. We booked the Turkish Bath through Alina travel (beside the mosque). £8 for the works plus a 30 minute oil massage. was a bit wary but would highly recommend for a bit of fun. My husband & I ( sounds like the queen?!) were on the marble slab together so I was reassured!!
Picked up& left back to our hotel & were taken to a hotel in Gumbet which had a Hamman.
Had a few extra treatments which weren't that great but no doubt how they make their profit.
Go for it!