Multiple Domain Question
You have a lot of options.
Option one is to tell the hosting company that you want both domain names pointing at the same web site. Some people believe that having your site mirrored in this fashion will hurt your search engine ratings.
Option two is to have one of the domains redirect to the other. This is the better choice if you prefer your new domain and want to use it in your branding. If this is the case then your web site should be moved to www.PreferedDomainName.com . Your old domain should be set up to redirect to the corresponding page on the new domain. For instance anyone who goes to www.oldDomainName.com/pagewhatever.html should be automatically redirected to www.PreferedDomainName.com/pagewhatever.html
Option three is to rebuild your web site to better optimized for search engines. In this case you would build a list of how your old pages correspond to your new pages. You would put the new site on www.PreferedDomainName.com . The old domain would use the list so that someone who goes to www.oldDomainName.com/pagewhatever.htm*l would end up at www.PreferedDomainName.com/correspondingNewPage.html*
Option four is to run both web sites as independent sites. Leave the old content as it is, and put your optimized site on the new domain. After a few weeks check to see which is doing better in the search engines, and stick with that.
In any case, you should use one domain for all your advertising and your emails. If you are changing the domain used for emails, you need to make sure that email sent to the old domain still gets delivered.
All of the above are relatively easily to implement. Depending on how you have farmed out the job, it will involve some combination of your hosting company and your web designers.
I run a small hosting company on the side, we have provided the above services to our clients. None of the above should be difficult for someone in the hosting business.