I'm not sure if these are coccineas or more likely, a microphylla. I would definitely go with the warm sunny room with the gro lites because they do not have roots. Definitely remove the flowers, since they will sap the energy that should go to root development

Do not root in water, but use a sterile porous and damp rooting media. Bottom heat will speed up rooting. When they have been potted up and growing has resumed, then they can go into the cooler room

rich d, Thanks for taking the time to reply.

Have to say that I have some pineapple sage sitting in a glass of water on my kitchen window sill with a very extensive root system. And a couple of flowers. Hope to get it potted up - maybe today? The Golden Delicious pineapple sage, that I rooted in a glass of water and then potted, is now a very substantial plant.