Nay, you do not need the expansion to play the Shadows of Angmar content and classes. Although I would recommend getting it as it unlocks Moria and other delights. The digital upgrade is quick and somewhat cheap.
If you used The Download Manager, I'd suggest totally uninstalling LotRO and TDM and download the full client directly and not through TDM. There were many problems with TDM and patching up to MoM, where the regular (CD/DVD or Digital Download) Installation and patcher was almost flawless, with only isolated hiccups. Best to just skip past those patches, and grab the latest full client from the downloads.
Download links can be found either at
http://www.lotro.com (hover over Support then cleft click Download LOTRO) or head to
http://lotro.stratics.com and look for the client download graphics on the bottom of the right-hand menu.
Welcome back to Middle-Earth!