Is this sentence correct?
It's nearly correct. There are multiple ways to fix the sentence.
Option 1: Switch to "nor".
"I don't want to go out, nor am I interested in going to your place."
"Neither do I wish to go out, nor am I interested in going to your place."
Option 2: Add a conjunction such as "and" or "but". In this particular case the result is clunky so I would not recommend it.
- "I don't want to go out, but neither am I interested in going to your place."
Option 3: Add a period.
- "I don't want to go out. Neither am I interested in going to your place."
Consider that "nor" is a negation of "or", and "neither" is a negation of "either". Formulating the whole sentence in the positive sense first could clarify the grammar:
"I (either) want to go out, or I'm interested in going to your place.."
Depending on the context, the usage of the "neither" and "nor" may be in an overly literary style. How could you say this in ordinary conversation?
"I don't want to go out. And I'm not interested in going to your place either."