There are several companies that try to have people renew domain names even though they are part of another Domain Name registar. I originally bought a domain name in 2001 for a nwo defunct web site and within weeks of buying the domain, I got a letter in the mail from Domain Registar of America or something like that, telling me I should renew....
In fact, just after I renewed my current domain name and transfered registration to Dreamhost, I had them snail-mail me again telling me it was time to renew and showing me my old expiration date on the domain name.
I would ignore network solutions if everything has been transfered to dreamhost... but transfering the domain name form your previous host to Dreamhost is not transfering the domain name from one registar to another.
I just checked your domain name on betterwhois.com and it looks like you are still with your old registar. You need to transfer it over. I can't give the step-by.step process of what to do off the top of my head, I hope someone else on here can do that for you.