Bracing 2x6 floor joists
2x6 joists at the span that you are talking about (16' and 18') are not strong enough to support a floor as a living space. With a span like that the timber size that you sister in would have to be at least 2x10's. Look this up in a joist span table (available in numerous places on the web) if you do not believe me.

Trying to install cross bracing on the existing 2x6's can stiffen the overall construction if it is installed correctly. It will not however make the overall floor any stronger. Instead it just spreads localized loads across multiple joists to lessen flexure due to live loads moving around. Depending on the method used it will add to the dead weight of the existing structure. If you tried to install cross braces using 1x3 or 1x4 material in an X format you would have to remove the ceiling from below in order to properly nail in the bottom ends of the cross braces. If instead the bracing would be done by installing 2x6 blocking between all the exiting joists these could be nailed from above using the appropriate tools. Although nailing 16d spikes in between 16" on center joists with a conventional hammer is no picnic. It is also likely that nailing to the existing joists like this will lead to some flexure and corresponding ceiling damage below.

It would be typical that a remodel like that would have, at a minimum, the roofing and roof sheathing removed on one slope of the roof so that the long members can be poked into the space from the outside. If the whole attic was basically clear it may be possible to make one hole in a gable end to push in the timbers and then position them in place in the attic space but it would be a total pain. 

Then with that all said you would have to decide if the loss of an additional four inches of floor to ceiling height under a sloped roof area ends up making a usable bedroom living space anyway. 

Rooms built up under a sloped roof often face a huge compromise of insulation in the rafter area where the sloped part of the ceiling comes into play. Done blindly with the idea of just thinking of attaching sheet rock to the bottom if the rafters you lose even more insulation due to the need to provide air vent paths in this space from the rafter tail area up to the peak area. Be prepared for a lot of heat loss in winter and the rooms potentially getting very hot in the summer.

A better choice remodel may be to consider an addition to extend the existing floor levels out the amount needed to provide for the added rooms. Or if there is an existing attached garage consider a garage conversion.