Your suspicions are indeed correct.
Most, if not all, dual SIM phones (not just WP, but Android and the simpler, more basic, Nokias (101, 107, 108, etc.)) tend to use (roughly) about 19% to 30% more power, when the second SIM is inserted. This is because the phone has to poll twice as often to the local phone transmitter. If you are in an area that has poor reception for one or, worse, both of the cards then battery drain will be considerably greater as the Phone searches for a signal.
If you switch to GSM, as opposed to 3G for either, or both, of the cards then that will help reduce the battery drain, and hence extend the life of a single battery charge.
As you suggest, turning off the second SIM (if it is not needed) will save power use.
There doesn't appear to be any official Manufacturer (empirical) data about the increased power consumption. I would have expected it to be in the specifications. Maybe it is something that they don't want to advertise, or bring to people's attention..?