The Oxford English Dictionary (OED) says either "cringy" or "cringey" are acceptable. Being that the OED is the ultimate source, the bible of the English language, I shall not doubt.
That said and being given the option, I spell it "cringy," not "cringey."
I base doing so on the fact that when something has a lot of dinge, it's "dingy." Now "dingy" pronounced with a hard G can mean a small boat or a ditzy person, but when it's pronounced with a soft G, it describes something as dirty or dirty-looking.
Another word that supports not adding the E is "spongy." The adjective form of "sponge" is "spongy," not "spongey."
I can think of no "vowel plus NG where the G is soft" orthography that supports spelling it with an E. I can only imagine that spelling it with an E arose out of it previously being spelled "cringe-y." Adding the suffix "-y" to the back end of an English root word is an accepted way to turn it into an adjective ad hoc, so until the word began to be used enough that it found its way into the dictionary, it was spelled "cringe-y."