Well, luckily a server load of right around 2 (provided it is a fairly high-end server) is not all that serious, but it looks like there is still a problem.
First of all, cPanel defaults to any server load over 1.00 as high and you will get an alert. 1 is not critical, and infact, probably isn't even enough of a load to really show any signs of performance degredation.
What is probably happening is your log files are just getting huge and bogging your server down when stats are processing.
If you have a dual processor server, you may want to consider this:
search for extracpus= and change the value from 0 to 2 or even 3.
save, close and execute /usr/local/cpanel/startup
This will raise your server load threshold so that your stats won't stop running with a measely 1.00 server load.
Second, while that will help eliminate the emails and the stats from not processing, I would recommend also disabling AWstats if you do not use them. I don't find awstats very useful at all compared to webalizer or urchin, so there is no need to process awstats if you don't use them right?
You can disable awstats globablly in WHM and I think even indivudally per account as well.