Thank you for sharing your situation with the Wefugees Community!
I am not familiar with the health system for refugees in Sachsen, but I called a migration counselling office in Leipzig to get some more information. They told me, that nowadays you unfortunately need to be in the asylum procedure for at least 18 months to receive an insurance card (the 15 months you were mentioning seem to be outdated).
Making some further research, I found a contract between the local authorities in Sachsen and the "Kassenärztliche Vereinigung Sachsen (KZVS)", which is kind of a union of doctors in Sachsen. According to section 3 of this contract (§3), if a dental treatment is urgently needed, asylum-seekers can choose between in the KZVS registered dentists with "orientation to the residential restrictions of asylum-seekers". Considering these phrases and the fact, that the health systems for refugees vary in Germany's federal states, I am afraid that you will get problems if you want to get a treatment outside of your "Landkreis" in which you are registered.
I would highly recommend you to see a doctor in Sachsen as soon as possible. I hope this helps and please feel free to get back to us with any further questions.
All the best,