I'm not a fan of buying plants at WalMart, but a pot is a pot as long as it has enough drainage. Culprit is more likely the soil than the pot.
How big are the pots relative to the plants? If you have very small plants in very large pots, it can contribute to the situation you described, since the plant doesn't use up very much of the water you put in to wet all that soil.
These are indoors, are the pots sitting in saucers? Don't leave water standing in them. And you probably don't need to water as often as you think. I only water my indoor plants twice a month.
It's still pretty amazing to me, south facing window and all, that you have a rose bush blooming indoors. If you had asked me, I would have said it would not work....