Hi, Good choice of places but between the two there is more to Rajasthan than Dharamsala. Also Oct-Feb is desirable time to go to Rajasthan. Having said that I suggest you cover one circuit properly. Choose between Jaisalmer/Jodhpur/Jaipur or Jaipur/Udaipur/MtAbu/Kumbalgarh/Ranakpur or Ranthambore/
Bharatpur if you have interest in wildlife. I would recommend 1&2 in view of 10 days available.Think about it. Good luck!
It has become a comparison between royal enfield and sports bike. Both the states keep their place in their place. If you want to get lost in nature then go to Himachal and visit Rajasthan, if you want to get connected to the royal chic, fort, culture, Thar....
Solely depends on your liking amd weather.
If its october or November then I can suggest Rajasthan... Jaipur- Pushkar( Pushkar Fair)-Jaisalmer
If u r heading to Himachal... U can visit Mecleodganj via Dalhousie