Hi Paul 2X6's 24" OC and spanning 16' will DEFINITELY require upgrading to work as floor joists. It sounds like you have balloon framing, where the 2X6's attach directly to the studs, with the subfloor holding the joist/stud assemblies apart. I would be inclined to put a ledger under the existing 2X6's to support the added weight of the 2X10's you plan on sistering. A 2X4 attached flat to the front of each of the existing wall 2X4's that goes up under this ledger will add much needed support and stability to this framing, and will allow the wall with your ledger to be finished, if you wish.
One other point....what is supporting this additional weight...i.e., what is under the weight bearing walls? Adding a lot of weight to the new floor above may be too much for this wall and its support. I'd check it out first.