If $f$ is continuous and $\frac{\partial f}{\partial y}$ exists and is uniformly bounded on a convex region $\Omega$, then $f$ is uniformly Lipschitz on $\Omega$. To see this, apply the mean value theorem at points $(x,y_1), (x,y_2)\in\Omega$, with $y_1$$\frac{\partial f}{\partial y}(x,y_3)=\frac{f(x,y_2)-f(x,y_1)}{y_2-y_1}.$$ If the partial derivative is uniformly bounded on $\Omega$ by $M$, we can then say that $$\frac{|f(x,y_2)-f(x,y_1)|}{|y_2-y_1|}\leq M\quad\hbox{for all}\quad (x,y_1), (x,y_2)\in\Omega.$$
Consider your example $$f(x,y)=x\sin y,$$ and lets look at this on the rectangle $$\Omega=\{(x,y): -a\leq x\leq a, -b\leq y\leq b\}$$ where $a$ and $b$ are positive. We have $$\frac{\partial f}{\partial y} = x\cos y$$ and so $$\left|\frac{\partial f}{\partial y}\right| = |x\cos y|\leq |x| \leq a \quad \hbox{for all}\quad (x,y)\in\Omega.$$ Thus $\frac{\partial f}{\partial y}$ is uniformly bounded by $M=a$ on $\Omega$.
(Definition: The function $g$ is uniformly bounded on the set $A$ if there is a number $K$ such that $|g(p)|\leq K$ for all points $p\in A$.)
A set $\Omega$ is convex if for every pair of points $p,q$ in the set, the line joining $p$ and $q$ is also contained in the set. In $\mathbb{R}^2$, examples include rectangles, parallelograms, discs, ellipses and $\mathbb{R}^2$ itself - but not stars or a disc with a bite taken out of it.