Learn English – “Neither….nor” or just ‘..nor’

Neither of your proposed options sound correct. Neither and nor are often used together, for example:

He ate neither meat nor bread for supper.

The meaning of which is

He did not eat meat for supper, and he also did not eat bread for supper.

Now, in the above examples, the neither/nor construction is idiomatic because you're referring to two things which he did not eat for supper. That is

He ate [x] for supper.

He ate [vegetables] for supper.

He ate [neither meat nor bread] for supper.

As you can see, both of the items referred to by neither and nor are things which were not eaten for supper. So it makes logical sense to pair them this way. (I'm not saying this is the only way in which neither/nor can be used, but it's common, and hopefully will help you understand the problem with your sentence.)

So now we return to your examples:

I am not your slave, nor do I have to follow your advice.

I am not your slave, neither do I have to follow your advice.

Neither of these sound right, and here's why: not being a slave and not having to follow someone's advice are two completely separate ideas. If they were both things I am not (your slave), things I don't have to do (follow your advice), or, as mentioned previously, things I didn't have for supper, then you could combine them with neither/nor. For example:

I am neither your slave nor your pet.

The category is things I am not; I am not your slave and I am not your pet. This is fine.

Now, it is also possible to change I do not have to follow your advice (things I do not have to do) into I am not required to follow your advice (which now makes it a thing I am not). In that case you could use neither/nor:

I am neither your slave nor required to follow your advice.

Now the construction makes sense because both items fall into the same category. This is actually a useable sentence for your purposes; you could say it and you would be understood just fine. It does come across as a bit formal, though, and I don't think you're likely to hear it in casual conversation.

So to select the wording which is most likely to be useful in conversation, forego neither/nor entirely since your two items are not of the same type. Instead say this:

I am not your slave and I don't have to follow your advice.

The and combines two separate but true statements. Simple and to the point!

You don't usually use "both" in the negative. Instead, you should use neither such as "Neither of them went". I don't mean that " Both of them didn't go" is not correct. What I mean is that the use of neither is more common than and preferable to the use of both" in this negative sentence. See the last paragraph of the link.