Anna - One thing you want to avoid is to make him feel obliged to dance with you. People are more resistant when they are obliged to do something. Score is zero out of four...and if it does not work next time, well, there are other matches and players in the future...move on. I think you are lucky because you have a husband that you can practice with anytime. Use him. Our instructors seldom dance with us during socials but I have a wife-partner to practice with all the time. After a while we are pretty much holding our own and have little need for our instructors during socials.
To avoid getting dizzy, avoid looking down on the floor as Len mentioned and do some spotting. There are probably more advanced dancers who can tell you more about spotting. Also, this website has a good description of spotting.
In my experience it is not uncommon for a woman to ask a man to dance, however, necessary permissions are to be obtained if they are with "significant" others. Again, this observation is only within the confines of studio socials or company of ballroom dancers. The conventional social etiquette of "man asking woman to dance" still holds true outside these settings.
Whatever happens enjoy and have fun dancing!