I know that for a compact Riemann surface of genus $g$, there should be $g$ different globally defined holomorphic 1-forms. However, I am having trouble finding any in my case. Here is the set up:
Let $M$ be a compact Riemann surface. Suppose that there are coordinate domains $U_z$ and $U_w$ both identified with the unit disk in $\mathbb{C}$ such that on $U_z\cap U_w$, $z=-\frac{1}{w}$. Then, does there exist a holomorphic 1-form defined on $U_z$ which extends to a 1-form on $U_w$?
More generally, are there any good resources for learning more about this type of construction?
Under the assumption that $z=-1/w$, you should be able to deduce that no holomorphic $1$-form on $U_z$ can be extended to $U_w$. Indeed, suppose $\eta=f(z)dz$ for some holomorphic function $f:U_z\to\mathbb{C}$. Write $f(z)=\sum c_kz^k$. Then the change of coordinates $z\mapsto -1/w$ takes $\eta\mapsto f(-1/w)d(-1/w)=\frac{-1}{w^2}\sum c_k\frac{1}{w^k}$ which is not holomorphic at $0$ (unless $\eta=0$). This should not be surprising news to you. The coordinate transformation $z\mapsto 1/w$ is the one which corresponds to $\mathbb{CP}^1$ and its standard open cover, which consists of the two hemispheres. Since $g(\mathbb{CP}^1)=0$, you know that there indeed aren't any non-trivial holomorphic $1$-forms on $\mathbb{CP}^1$.
Fermat curves give us examples where it's possible to explicitly compute a basis for $\Gamma(X,\Omega^1_X)$. The notation below will be slightly different from what you are using, because I've copy pasted this from some notes I wrote a while ago. The notation $Z(f)$ denotes the vanishing locus of the function $f$.
Consider the Fermat curve $X=Z(U^n+V^n-W^n)\subset\mathbb{CP}^2$. On the affine patch $D(W)$ (i.e. where $W\neq 0$), take coordinates $u=U/W$ and $v=V/W$. Define a $1$-form on the open subset $D(u)$ where $u\neq 0$ by $\frac{dv}{nu^{n-1}}$. The affine curve $D(W)$ is defined as $Z(u^n+v^n-1)$, so $d(u^n)+d(v^n)=0$, meaning $nu^{n-1}du=-nv^{n-1}dv$. On the intersection $D(u)\cap D(v)$, we then get $$\frac{dv}{nu^{n-1}}=\frac{-du}{nv^{n-1}}$$ Clearly, the latter extends to $D(v)$ holomorphically. Since $D(W)=D(u)\cup D(v)$, this means we can glue the two to get a holomorphic $1$-form on $D(W)$. We want to extend this to $D(W)\cup D(U)=X$. Choose coordinates $x=V/U$ and $y=W/U$ on $D(U)$. Then $D(U)=Z(1+x^n-y^n)$. On the intersection $D(U)\cap D(W)$, we have $v=x/y$ and $u=1/y$. This leads to \begin{align*} dv=d(x/y)=\frac{dx}{y}-\frac{xdy}{y^2}\implies \frac{dv}{nu^{n-1}}=\frac{y^{n-2}dx}{n}-\frac{xy^{n-3}dy}{n} \end{align*} Assuming that $n\geq 3$, this can evidently be extended holomorphically to all of $D(U)$. Thus, we can glue the $1$-forms on $D(U)$ and $D(W)$ to get a non-constant $1$-form $\eta\in\Omega^1_X(X)$.
In fact, we can produce more $1$-forms on the Fermat curve by a similar argument. Consider the $1$-form $$\frac{v^kdv}{nu^m}\in\Omega^1_X(D(u))$$ Using that $dv=\frac{u^{n-1}du}{v^{n-1}}$ again, we see that on the intersection $D(u)\cap D(v)$, we may write $$\frac{v^kdv}{nu^m}=\frac{v^{k-n+1}du}{u^{m-n+1}}\in\Omega^1_X(D(u)\cap D(v))$$ This expression can be holomorphically extended to $D(v)$ if and only if $0\leq m\leq n-1$. Suppose this holds, and we want to extend to all of $X$. The same relations between $x,y,u,v$ on $D(U)\cap D(W)$ as before yield \begin{align*} \frac{v^kdv}{nu^m}=\frac{(x/y)^k}{n(1/y)^m}(\frac{dx}{y}-\frac{xdy}{y^2})=\frac{x^ky^{m-k}}{n}(\frac{dx}{y}-\frac{xdy}{y^2})=\frac{x^ky^{m-k-1}}{n}dx-\frac{x^{k+1}y^{m-k-2}}{n}dy \end{align*} Once again, these may be extended to all of $D(W)$ if and only if $0\leq k\leq m-2$. Thus, we get $(n-1)(n-2)/2=g(X)$ choices of $m$ and $k$ that yield a holomorphic $1$-form on $X$. Using Riemann-Roch, one can in fact show that $\Gamma(X,\Omega^1_X)$ is a finite dimensional complex vector space of dimension $g(X)$, which means that we have produced a basis for $\Gamma(X,\Omega^1_X)$ (at least, after you show that they are linearly independent over $\mathbb{C}$).
The above example illustrates that the precise form in which your Riemann surface appears is quite important. It's not just a topological surface, it also has a complex structure, and you need to specify this complex structure before you can perform the calculation that you are asking about. For more information, I would refer to Rick Miranda's book "Algebraic Curves and Riemann Surfaces".