CPanel Dedicated Server Nameserver Issues

Hi -

My company has a dedicated server running CPanel that I have cursory involvement in - but only insofar as if something messes up it ends up falling into my lap to fix.  CPanel isn't my admin tool of choice - but I'm comfortable enough to navigate around in it.

The server is messing up - insofar as at some point - it gets bogged down (I'm assuming a ram issue?) and the nameserver functions start messing up (sites cease resolving).  So - I can't get to the server if I use the actual domain names - but I can get to it if I just use the direct IP of the server.

Now - I've got a couple of suspicions as to what's causing it - and I'd like your perspectives to see if I'm off base or if I'm probably on target.

Most likely cause (in my opinion)
Too many open/sleeping MySQL connections  - There's a site that exists off-server that communicates with the SQL server on ours.  When I look at the open MySQL processes - at some point it bogs up and there are HUNDREDS of open/sleeping MySQL processes going which have been open/sleeping for hours and hours.  

I'm thinking that the other service doesn't close the connections properly after each transaction - and eventually that just bogs down the server.

Secondary possible cause (in my opinion):
the /dev/sda1 partion is too full and there's too much memory swapping going on

/dev/sda1             9.7G  6.4G  2.8G  70% /

What do you think? Is this something you've run into before - and do my guesses sound on target?  Or is there something I might be missing.

Thanks!