‘Don’t’ is a contraction of ‘do not’, and ‘do’ is a verb meaning ‘to perform/execute’. Strictly speaking, then, are these two common constructions ungrammatical?
a) ‘Don’t do this/that.’ Since it expands to ‘Do not do this/that’, isn’t  the second ‘do’ redundant? If so, why does ‘Don’t this/that’ sound so wrong whereas the expanded ‘Do not this/that’ merely sounds a bit archaic?
b) ‘Don’t continue.’  Since both ‘do’ and ‘continue’ are verbs, shouldn’t it be ‘Continue not’? 
 ‘Isn’t’ seems to follow this remain-grammatical-albeit-archaic-sounding-when-expended rule, as does ‘shouldn’t’ in question (b).
 Ditto for any other verb in place of ‘continue’.
 And if you wanted everyone to think you were odd you could presumably insist on writing ‘continuen’t’.
The answer is that English doesn’t need to use “do not + nonauxiliary*” (aka “do support”), since it did perfectly well when it didn’t. It’s just that you need to use it in order to not sound archaic.
For example, the Tyndale Bible, 1526:
Yf I do not the workes off my father, beleue me not.
The reason it sounds archaic is because it is. English had other types of negation besides “verb not” and do support, but they were either too old or too uncommon, so they no longer sound like grammatical archaic English. I covered these forms in a previous answer here (this previous answer also explains why we use do support instead of other types of negation).
As for why it doesn’t sound right to try and contract not without do support, I think this is partly because of rarity and partly because contractions don’t sound like archaic English, and are thus not used when trying to sound archaic. You can find a contraction of “know not” in Wuthering Heights:
“Well, for sure case, I knawn’t how they can understand t’one t’other: and if either o’ ye went there, ye could tell what they said, I guess?”
*Some words are both auxiliaries and nonauxiliaries. “Need”, for example, is one that can be used as an auxiliary or nonauxiliary pretty much anywhere, but do is only an auxiliary in certain situations: “don’t do” has both an auxiliary and nonauxiliary do.