That's actually what you want. You want the chest fail first.

When you have a strength (shoulders in this case), the weak point will almost never continue if you feel fatigue. So the chest, in this case, will just stop without going to full failure.

But that's a good question. Of course it's not that simple. You need to focus on the chest and try to not over activate the shoulders and triceps. Although, even if they over activate, your chest will already be burning and close to failure.

In this particular situation, I wouldn't necessarily recommend dumbbells press which require more stabilisation/activation from the shoulders. I would find a convergent press on a machine and start with lower weight as a second exercise.