Suppose you are writing and you are tempted to put this down: "He quickly ran..." I counsel against doing so. It dilutes the impact of your words. If you write "He dashed..." or "He sprinted..." or something else, you use a more dramatic, evocative verb. And that's a good thing.
Maybe in an early draft, you can use weak verbs with adverbs to just get that part of the story onto the paper. But remember to go through your text again with find function of your word processor looking for "ly" and every time you run into an adverb, ask yourself? Can I get rid of this?
What this will do is force you to rack your brain for better verbs.
It will also force you to think about what you're doing in the scene. Maybe the adverbs are there because you're telling, and not showing. Same for all modifiers: adjectives and prepositional phrases. These are little tokens that you've interpreted the scene and that's a task best left to the reader as I've mentioned elsewhere.