Learn English – Proverb to explain the given situation

The most common English phrase for this is between a rock and a hard place. It means:

In difficulty, faced with a choice between two unsatisfactory options.

Specifically, to say you are caught between a rock and a hard place means that you are in a dilemma. The exact dilemma seems to be exactly what you're describing--Morton's fork, which is:

a choice between two equally unpleasant alternatives (in other words, a dilemma), or two lines of reasoning that lead to the same unpleasant conclusion. It is analogous to the expressions "between the devil and the deep blue sea," "between a rock and a hard place," or, as those in the Spanish- and Portuguese-speaking world say, "Between a cross and a sword." This is the opposite of the Buridan's Ass.

Here is a related line from Shakespeare that has become idiomatic:

The lady doth protest too much, methinks.

It comes from Hamlet and you can read all about it here. As the Wikipedia article suggests, it has come to mean that one can "insist so passionately about something not being true that people suspect the opposite of what one is saying."

I have heard it in many forms, such as "methinks the lady doth protest too much," "the lady protesteth too much," or more simply "she/he protests too much."