It seems that your DM has the magic dead-zones there for a reason.
For one, these dead zones may not function in the same way as the 8th-level spell Antimagic Field. You need to ask your DM or find out as you adventure further into the campaign.
There is some information for antimagic field effects for Dead Magic zones (DMG p.109). If the these zones work in the same way, you do not have many options as per this previous post:
How to get rid of Anti-Magic Fields?
However, even in this scenario, you could work try to work out a warning system to help you manage your way through a magic dead-zone. This way at least you have fair warning and don't find yourself in the midst of mob firing blanks!
For instance, let's suppose the dead zones are of an irregular shape and they do not cover the whole area you are in. But how can you know? In this case you might use a continual spell-effect on an object to serve as a warning to signal that you are going into a dead-zone. If you try casting a Cantrip you will find out without expending a spell slot, but that will use up your action, but this will get very tiresome.
You could use the 2nd-level spell Continual Flame on an object. For example, casting it on a fishing lure on the end of a very long fishing rod. Then just hold it out in front of you. When the lure hits the dead-zone it will repress the Continual Flame. Once you are back out of the zone it will light up again. :) ...an antimagic radar of sorts.
You could get more creative with this. Possibly train an animal companion wearing a collar with Continual Flame cast on it. You could cast it on an arrow and shoot it across a long distance to test the area, and so on.
But again, go back to the principle of why your DM has these zones in your campaign, respond to them creatively. See it as a challenge and have fun!
I hope this helps.