You will get your visa in Istanbul whether you are leaving the airport there or going onto
DalamanIf the flights were all booked online or via an agent in one transaction then you do not have to pick up your luggage and it will go onto your connecting flight
If you have an 11 hour wait and it is in the day then you could take the Metro below Ataturk airport and go along to Bakirkoy where you will find many shops, bars, cafes and lots to see and do and then get back to the airport in plenty of time for your next flight
Don't forget the Dalaman flight is classed as a domestic flight so you can walk in an hour before to take this one not like the international flight where you are there hours before
Take the Metro and go and soak up the atmosphere in Bakirkoy and do some people watching
here is a link to the Metro service available so to cut out all the traffic
http://www.istanbul-ulasim.com.tr/media ... -01-01.jpgIf the roads were clear then you are talking a 10-15 minute drive anyway and I have been there many times myself
Thanks for the link Rami