I think your usage, "I hear it a lot while watching cartoons ...", is correct. The word cartoon is a countable noun. A singular countable noun will need an article (i.e. a cartoon or the cartoon) or a determiner (e.g. that cartoon). Thus, we can say watching a cartoon or watching cartoons, but not *watching cartoon.
However, you observation made me wonder if "watch cartoon" had become a phrase (as "declare war" did) already.
So I looked into some of the examples returned by Google Ngram, and found that all the "watching cartoon" results (as far as I reviewed) are not a valid noun phrase. For example, All your little friends were watching cartoon shows ..., ... watching cartoon turtles on TV doing ..., ... that children watching cartoon shows for a four-hour period .... In all results, the cartoon was used as an adjective.
With that, I think I can conclude that saying *watching cartoon is incorrect.
