"Not only..., but (also)..." is used to create a certain kind of emphasis. Let's look at some examples:
Not only was she an accomplished pianist, but she could also write great poetry.
In this example, we have a person who is already really good at something, but wait, there's more! She is also really good at something else.
Not only did I pass the test, but I also got the highest score in the class.
Again, passing the test is just one part of the story, but there's more - the speaker did better than everyone else. In both cases, the second part of the "story" is something that's not obvious or implied by the first part.
Your example sounds like it's used for comic purposes, because eating and watching TV are everyday things and there is nothing unusual or unexpected about doing them at the same time.
While I was watching TV, I was eating dinner.
This is a simple statement that one thing was happening while another thing was also happening. Another example would be "While I was walking to work, I was thinking about my many projects."
While I was watching TV, I eat dinner.
This sentence is grammatically incorrect - you're mixing a past tense and a present tense, and in English the tenses of verbs have to be in agreement within the same sentence. You could say "While I was watching TV, I ate dinner", which means that you were watching TV long enough to start your dinner and finish it, and possibly kept watching TV after that, or started watching it before dinner. In other words, the emphasis isn't that watching and eating were happening at the same time, but rather that all the eating was done and finished while the watching was going on.