Finite State Machine to find Greatest Common Divisor
You actually have all written down, you just need a couple of registers and a little combinational logic. I'll start writing down how I think you can do the machine defining inputs and outputs while designing it.

Let's say you start in your \$S_0\$ state, that's your "entry" state. First things first, load the two registers. On the first transition you will need:
$$\begin{cases}
\textbf{LDA}=1 &amp; \textit{to write in A}\\
\textbf{LDB}=0 &amp; \textit{you don't want to write in B}\\
\overline{\textbf{W}}=0 &amp; \textit{to actually write}
\end{cases}$$
On the second one of course:
$$\begin{cases}
\textbf{LDB}=1 &amp; \textit{to write in B}\\
\textbf{LDA}=0 &amp; \textit{you don't want to write in A}\\
\overline{\textbf{W}}=0 &amp; \textit{to actually write}
\end{cases}$$

You now are in another state, remember that you passed through a "silent" state. The last two transitions are conditionless, i.e. they do not depend on the inputs.

Let's call your current state \$S_L\$, L as in loop. We are now implementing the Euclid's algorithm just as you wrote it down. Firs of all the easiest thing: when A=B we need to exit the loop and tell the outside world we are done, so another transition would be:
$$\textbf{if }\textbf{CC}=00 \rightarrow S_{end},\ \textbf{F}=1,\ \textbf{LDA}=0,\ \textbf{LDB}=0$$
There are two more transitions of interest, the important input of course is CC:
$$\textbf{if }\textbf{CC}=01 \rightarrow S_{L}, \textbf{F}=0,\ \textbf{OP}=0,\ \textbf{W}=1,\ \textbf{LDA}=1,\ \textbf{LDB}=0$$
$$\textbf{if }\textbf{CC}=10 \rightarrow S_{L},\ \textbf{F}=0,\ \textbf{OP}=1,\ \textbf{W}=1,\ \textbf{LDA}=0,\ \textbf{LDB}=1$$
I'm calling "FOUND" just "F". All the outputs that are not specified in a transition can be treated as "don't cares"

Finally, about \$S_{end}\$, you might just want to stay in it keeping \$\textbf{F}=1\$.

Summing up:

States: \$S_0, S_1, S_L, S_{end}\$

Inputs: CC
Outputs: LDA, LDB, W, OP

You have all what you need to sinthesize the FSM, come back if you need help with that too.