None of these searches would have been lawful had the women not consented. The Fourth Amendment is there for a reason. There is absolutely no reason to waive such a fundamental and important protection. A few finer points should be noted. Consent need not be written to be valid.
Oral consent is perfectly valid.
It is a misconception to think that if you don't sign something your consent was invalid. In fact, in the bizarre reasoning of the U.S. Supreme Court, you can waive you right without even knowing you have a right to say no to the police. Consent searches are part of the standard investigatory techniques of law enforcement agencies. They normally occur on the highway, or in a person's home or office, and under informal and unstructured conditions. And, hundreds of times each day, the bluff pays off, the Constitution is skirted, and the sovereignty of citizens subverted.