Not quite. Dash usually implies running in the sense you're using, and specifically running a relatively short distance. Take, for example, the olympic events 100m dash, 200m dash, etc. (Though other meanings also exist for "dash"; e.g., a small amount of something in a recipe.)
So you might dash into a store, for example, if you were literally going to run in very quickly from across the parking lot. But even in that case, "dash" would be a bit uncommon. Probably just "run in" would be more typical.
But the phrase you're looking for is dive in. From Merriam-Webster:
to plunge into some matter or activity
Example sentence from MW:
she dove into her studies