Yes that fan on it is very important It's acting as a passive cooler if there isn't a fan on it and the effectivness of it is greatly reduced. Essentially the dead fan is blocking airflow to the heatsink too so that's making it worse.Try giving the fan a little push to see if you can get it started again (not while it's spinning of course )
Normally I would always go with zalman for heatsinks but the thermalright is also a good performer Its mainly up to you and whether price is an issue or not?
:edit: edited for spelling and punctuation