Your option (a) is usually used like this:
In spite of all my efforts, I couldn't pass the final exam. And now I am back to square one! The same old story!
Example from Cambridge: If this doesn't work we're back to square one.
We say "I am [back to square one]", "you are", "he is", "they are", "[he had] to go [back to square one]". MW gives the following example:
His idea didn't work, so he had to go back to square one.
"..back at square one" is also correct, depending on the context.
The two idioms can be used interchangeably in most cases. "Back to square one" is more often used than "back to the drawing board". But there are circumstances when one may be more preferred over the other.
To me, "back to the drawing board" seems more appropriate for creative endeavors (rethinking one's strategy/approach to tacking a problem, brainstorming, designing a plan differently, trying a different way to teach when students appear confused, etc). But I can't find any credible sources to substantiate my opinion.
Here are 3 extreme examples:
Example 1: You spent 5 hours writing an essay or writing codes for your assignment. But you forgot to "save" your work. For whatever reason, the power went out and you lost everything. You would say "I am back to square one" meaning that you have nothing now and that you have to start again.
Example 2: You made a numerical mistake in one of your first order conditions (calculus) and followed through. You interpret your results in detail and make a recommendation. But then someone points out your mathematical error, and you realize you have to start over. I would use "back to square one", but not "back to the drawing board" because the error was a calculation mistake; the interpretations and recommendation based on the faulty number was correct. The strategy (steps) to tackle the problem was correct. You don't have to change the way you approached the problem.
Example 3: You are taking a road-test for your driver's license. You drive around for 2 hours; all is good. You panic while trying to parallel park and slightly graze a parked car. You fail the test; you have to take the test again but you have to wait for another 4 weeks before you can. This is a very simple issue. You don't have to rethink your strategy completely. You already have the driving skills. You just need to practice and not panic. In this case, I would use "back to square one" and not "back to the drawing board".
I think "back to square one" can be used in every situation, but "back to the drawing board" is more appropriate for situations where one needs to change their approach/strategy (i.e., go back to white board, rethink, and try a different approach).
Edit: You can be "back at square one" for another person's screw up. You apply for admission to your favorite college. But your high-school administration got swamped with work and failed to send in your transcripts on time (or just forgot completely). You don't get in and need to reapply next fall. You are "back to square one".
