Hi Emarbe,
We use "badly" to emphasise on the seriousness of an action. It has a negative connotation. Yes, it can be used with some verbs.
Example: .... miss you badly, .... badly burnt, ..... behaved badly.
On the other hand, we use "very much" to mean that something is happening to a great extent.
Example: I miss you very much, I would very much like to attend the concert, Thank you very much, ... enjoyed ourselves very much.
Not every verb can go with "badly" or "very much". For example, we cannot say: .... rained badly (rained heavily, ..... apologised very much (though we can say thank you very much). Some words naturally collocate in English.
Hope this helps.
Cheers, Lance