What prevents a toilet auger from fully extending into the toilet

If the problem is mineral clogging of the under-rim channels on the toilet, and this does produce the symptoms you describe, then neither CLR no vinegar may be enough to clear up the blockage. The traditional treatment is a cup or two of muriatic acid (HCl 36%, from the hardware store) down the overflow hole in the tank. That goes directly to the rim vents. Pour the stuff in, wait an hour, and flush. Gloves and eye protection are recommmended, and a funnel so as to avoid spillage and corroding the metal parts in your tank. This looks like decent writeup of the method: How to Fix a Slow Toilet

If the sink is backing up into the toilet, then the clog is at or after the junction of the two in your drain lines. You'll have to snake the drain to remove the clog, and it will likely be beyond the reach of a standard toilet auger. You could remove the trap on your sink and attempt to snake it from there, or remove the toilet and run the snake down that drain.

If you do remove the toilet, you should wait for the water in the lines to recede so it doesn't flood your floor. If you go through the sink drain, you'll likely have a smaller drain line to maneuver the snake and it's possible the clog is further away, requiring a longer snake.