Drains do not freeze because the are empty except when actually draining water. Your problem is a clogged drain, but you are finding what is often explained on this forum, DIY hand snakes are almost too puny to clear a clog. They usually just poke a hole through a clog but not clear it. Often they will just pack the clog tighter. So, the thing that you need to do is call a professional plumber with an industrial sized auger that will auger the drain clear out to the street. Do not rent one of these big units even if you can find one. They are extremely dangerous for the inexperienced and even if you escape injury, you could damage the unit and/or your plumbing. This is one job best left to the pros. Also, do not use chemicals.