This answer assumes that your husband has never lived in another EU country. If he has, then you may be able to settle under the EU freedom-of-movement regime, about which there is more information below.
Is it true that the German authorities will ask me to exit the country and apply for a family reunion visa, even though I'm gay, Yemeni and my residency in Turkey will expire in August?
I do not know much about the German settlement rules for non-EU family members of German citizens, so I encourage anyone who does to add another answer with that information. So I cannot answer this part of your question, but I will assume that the answer is "yes."
This leaves two options, one of which you've asked about:
1. Apply for asylum
I don't know much about this, but your claim has to be based on a fear of persecution in your country of citizenship. It appears that you have such a fear, so it seems like a legitimate option for you to consider. If you decide to pursue it, you should seek help from a lawyer who has experience with asylum claims. The last thing you want is for the claim to be rejected because of some technical deficiency in your application.
2. Settle under EU freedom of movement
You and your husband can go anywhere in the EU together, because, as his spouse, you are a "person enjoying freedom of movement under Union law" when you travel with him or travel to join him somewhere. This is controlled by directive 2004/38/EC; there are also a summary of the legislation and a public information page describing the situation.
The only place where you can't do this is Germany, because German citizens and their family fall under German national law rather than EU law. There ought to be an exception if your husband has exercised his freedom of movement by living elsewhere in the EU, based on the Surinder Singh case, but I cannot find any evidence on German government websites that Germany has implemented procedures to comply with this ruling.
You could use the second option as a fallback if your first option fails. You might want to talk to a lawyer to make sure you will be able to avoid deportation to Yemen so you can get over the German border into another EU country.
In theory, once you've lived elsewhere in the EU for six months or more, you should be able to move back to Germany under EU freedom of movement, but if Germany has indeed failed to take the Surinder Singh ruling into account, you may have trouble with this. I have asked a new question about this.