Hi American Mum,
I am an American living in Nottingham with my own relocation company and i can answer some of your questions. No, there is not a mulitple list system like there is in the US, however, it is not uncommon for a landlord to put his property with more than one agent. A letting agent will have no problem dealing with you without your husband present. Letting agents will charge an application fee when you decide on a property. Don't know if they do that in the States. Technically, the fee is to do a credit check on you which is impossible since you are coming from the States but it is how they make their money. Oh yes, another thing....most letting agents are useless, hopeless imbeciles who make my life a misery.
Much better to hire a relocation agent and let them do all the horrible leg work and negotiating for you. They can give area familiarisation tours, info on the best restaurants, places to park, hair dressers, dentists, doctors, schools, etc. etc. After doing this job for 10 years I can't believe there are people who relocate from other countries without one!
Good luck!