You're on the right track, the drain is clogged. However, you are finding out what is frequently discussed here that a DIY snake is seldom effective. I suspect what you have hit is a tee in the drain that is stopping the snake. Even if you get past that point, you are likely not to clear the blockage and may even push it in deeper and tighter. What is required is a professional with a professional size auger to really open the drain. Whatever you do, do not use caustic chemical cleaners. These not only won't open the drain, they can damage pipes and create a hazzard for a plumber who has to finally come in.