Here's an example with a topological action of $\mathbf{R}^2$ on the 2-torus with 3 orbits; I expect it can be made $C^\infty$ but definitely cannot be made real analytic.

If $X$ is a topological space and $f$ a self-homeomorphism, recall that its mapping cylinder $\mathrm{Cyl}(X,f)$ is $X\times [0,1]$ modulo identification $(x,1)\equiv (f(x),0)$. It admits a continuous action of $\mathbf{R}$ given by $t\cdot (x,u)=(f^{-\lfloor t+u\rfloor}(x),u+t-\lfloor u+t\rfloor)$. If $X$ is a smooth manifold and $f^{\pm 1}$ is smooth as well, then so is the mapping cylinder and the action. (Here smooth can mean $C^k$, $C^\infty$, $C^\omega$).

Now consider a continuous action of a topological group $H$ on $X$ commuting with $h$, in the sense that $h\cdot(f(x))=f(h\cdot x)$ for all $x\in X$ and $h\in H$. Then the action of $H$ extends to an action on $\mathrm{Cyl}(X,f)$, simply given by $h\cdot (x,u)=(h(x),u)$, and this action commutes with the flow, so that we get a continuous action of $G=H\times\mathbf{R}$ on $\mathrm{Cyl}(X,f)$. Clearly if the action of $H$ has finitely many orbits, then so does the action of $G$, and if the action of $H$ has locally closed orbits, then so does the action of $G$. Also if the action of both $f$ and $H$ is smooth in any sense then so is the action of $G$.

Now we pick $X$ to be the circle, identified to the projectivized line $\mathbf{R}\cup\{\infty\}$ by sterographic projection, $f(x)=x+1$, $f(\infty)=\infty$. Then we pick $H$ to be the reals, acting in such a way that its orbits are the integers (singletons) and the intervals between integers. [This action can obviously made analytic outside $\{\infty\}$ but behaves badly close to infty]. Anyway the action of $G=\mathbf{R}^2$ on the 2-torus $\mathrm{Cyl}(X,f)$ has 3 orbits: $C=\{\infty\}\times [0,1]$, the orbit $C'$ of $(0,0)$, which is orbit of $(0,0)$ by the flow of the mapping cylinder and is fixed by $H$; it's an immersed line, accumulating on the orbit $C$, and the dense open orbit $C''$, on which $G$ acts freely. Then $C'$ is 1-dimensional and so is the circle $\bar{C'}-C'=C$.

So far this is only a topological action of $\mathbf{R}^2$ on the 2-torus. To get a better smoothness, we need the action of $H=\mathbf{R}$ to be made smooth. To make it real analytic is hopeless because $\infty$ is an accumulation point of isolated fixed points. On the other hand, to make it $C^\infty$ does not sound hard.