The simple answer to your question is "Yes, all of them!" No problem in NP is in #P because NP is a class of decision problems (problems where the answer is yes or no: for example, is this graph 3-colourable?), but #P is a class of function problems (problems where the answer is a non-negative integer: for example, how many 3-colourings does this graph have?).
You seem to be confused about the basic definitions. A decision problem is in NP if there is a nondeterministic, polynomial-time Turing machine that has at least one accepting path whenever the answer is "yes" and no accepting paths whenever the answer is "no". A function problem $f$ is in #P exactly when there is a nondeterministic, polynomial-time Turing machine that has exactly $f(x)$ accepting paths for every input $x$. So, NP is about the existence of accepting paths, while #P is about the number of them.
I'm not sure what you mean by "if you already have a [...] Turing machine that can accept correct paths". Turing machines don't accept paths — they accept (or reject) their input. An accepting path of a Turing machine is a computation that it makes, that ends in an accepting state.
Here are two facts that you might have been thinking about when you asked your question.
- For every function $f$ in #P, the decision problem "Is $f(x)>0$?" is in NP. This is because the Turing machine that computes $f$ (in the sense that it has $f(x)$ accepting paths for every input $x$) has at least one accepting path (i.e., accepts its input in the sense of NP) exactly when $f(x)>0$.
- Every decision problem in NP is contained in P#P. That is, if you have an oracle for any #P-complete problem, you can use it to answer any NP decision problem. This is because every problem in NP is accepted by some Turing machine $M$. The machine $M$ computes some function $f$, where $f(x)$ is the number of accepting paths that $M$ has for input $x$. So, to find out if $M$ has at least one accepting path, you just use the oracle to compute $f(x)$ and then check if that value is greater than zero.