" how do you manage to keep the plants adequately watered considering the
sprinkler head I am using is tied into a zone in my sprinkler system
and is scheduled to run for 20 minutes three times a week."

Honestly I don't think you can. Not unless you can somehow create a new separate zone for the garden or find some other way to feed the drip system.

Such a frequent shallow watering approach may be fine for grass but not for most vegetables. Many of the common veggies are deep tap root plants and even those that aren't have a root mass 6-8" down.

Drip irrigation systems are design to run for much longer periods of time - think in terms of hours, not minutes even with the larger gph emitters. Such shallow watering as you describe only encourages shallow rooted, water dependent plants in the vegetable garden. Deep, infrequent watering is the ideal goal with the vegetable garden.

Dave