Can you drop your husbands insurance through his work and get your own insurance?

Yes.

The best way to go about doing this would be to get quotes before the annual period in which you can enroll/disenroll, and go ahead and apply for an effective date that is the same day that you can make changes. That way, if anything stops the private company from issuing the insurance you can keep the work coverage without having any period where you are without coverage.