I have hot AND cold water in the kitchen, cold water ONLY in the bathroom sinks and NO water at all in either bathrubs. When I first turned on the cold water, it came out a brown/rust color. Houston is experiencing record lows so I'm wondering if the water was temporarily shut off and this is initiated my problem. Please help.

Could be caused by water company working on the line stirring up debris. It does sound like debris so handle each one separately one by one. Remove all areators, remove and clean cartridges and/or stem depending on what you have. With stem or cartrdige removed place a tightly held towel over the opening and turn the supply back on to flush the line.