Learn English – Using “neither” without “nor”

No, "neither" must always come first in this construction.

Also it generally comes after the verb in the example you give, not before it:

I like neither thing A nor thing B.

As @Richard points out, putting "neither" before the verb implies that the verb is part of what the "neither" is applying to:

I neither like nor detest thing A.

Although the classic rule is to use neither and nor together, Grammar Girl writes:

“Nor” doesn’t necessarily have to appear in a sentence with the word “neither.” “Nor” can start a sentence. For example, if you’ve just mentioned that you don’t usually wake up at 6 a.m. and you want to continue being negative, you can start another sentence with “nor”: “Nor do I like to wake up at 5 a.m.”

Another option is to combine the two negative ideas into one sentence and then start the second part with “nor”: “I don’t usually wake up at 6 a.m., nor do I like to wake up at 5 a.m.”

In your case, you have three options. From the above, it is correct that you don't need to use neither. However, in the same article Grammar Girl writes:

You may also use “nor” if you’re talking about more than two items, but you must repeat “nor” after each element (2). So if you want to add ketchup to your list of dislikes, you have to say, “I like neither hot dogs nor mustard nor ketchup.” It would be incorrect to use an “or” anywhere in that sentence—or to leave out either case of “nor.”

So taking your list as even a single sentence, I think you need to repeat nor. So you could write (note grammar change in the second sentence):

I wasn't going to play dead.

Nor was I going to hide.

Nor was I going to run away.