Your question really needs a doctor's answer of course, and since I'm not a doctor I can't give you one.
What I can give you is that I am on medication for GAD, and what I take is one of the "SSRI" drugs. And yes, it makes me drowsy!
What I do is take my tablet (one a day) after dinner at night so the drowsiness is dealt with by my being asleep, and I am still medicated in the morning to face the day without wanting to shed blood.
The other key I suppose is that drowsiness in itself is a symptom of many Anxiety disorders so you might just feel like that because you are sick, not because the drug is doing it. Again, I'm not a doctor, but this has been my experience.
I've been on my tablets for about three and a half years and my Psych as suggested I actually increase my dose because Anxiety requires a higher dose than Depression. Of course I'll only do that with proper advice and after certain medical checks first, (so I'm not telling you to do it!!!!),.but if medication is part of the treatment it may just be something we both need to deal with.
I experience Anxiety and that makes me tired. It sucks, but it's true.