Adjustments for baking a layer cake in one pan?

Volume isn't really important in this sort of situation; the significant factor is how far the heat has to penetrate, so thickness is your main consideration; I agree with the suggestion to use a rectangular or otherwise shaped pan with similar surface area- for 2 8" layers the surface area would be almost exactly 100 sq.in., so a 10" square pan would be perfect, but it doesn't have to be exact, just something in that range(7x14 would be great as far as the shape of the final cake. If you decide to make a tall cake it may also have trouble supporting it's own weight and tend to collapse. If the recipe doesn't specify flour, it probably means AP, but recipe writers have their own standards- if it's in a book it's always a good idea to check the "ingredients" section for this sort of information.

Susan G. Purdy, in A Piece of Cake, has a section on comparative pan volumes. She indicates that the maximum capacity for an 8" pan is 2 cups of batter; since your recipe specifies two such pans, figure on 4 cups batter total. Other options for 4 cups batter include and 8x12 oblong if you have that; a 9" wide x 2-3/4" deep tube pan, or a 6-1/2" wide x 3-1/4" deep soufflé pan. I'm guessing you have the latter.
It's not possible to say what would happen with the pan that you have but I think it's worth a try, perhaps at a slightly lower temperature.
When my kids were small I baked a "Barbie" type cake, using a metal mixing bowl as the cake pan. Because it was so deep, it took much longer than expected, but it worked fine. I have occasionally used a Pyrex soufflé pan for brownies when I didn't have the pan I needed.
Cake baking depends on the materials used and their ability to retain heat as well as the pan size; if you have enough time, you can experiment with your pan to see if it works. Alternatives include the pans listed above and cupcake pans if you have them (you'd need the liners) or even disposables. You can get disposable aluminum or paper pans without investing a lot. But if the deep dish is all you have, I'd just try it. Just be aware that, as Nancy says, the crust will most likely cook before the interior is ready and you may well end up with a thicker crust than you would with a shallower metal pan.

I would worry about baking a 4-layer cake in a deep 6" pan...the middle might remain uncooked or the outside parts dry out before the whole thing was cooked.
You might consider a shallower rectangular pan, one large or two medium, in which to cook the whole recipe. Then cut out the 6 in layers (use cardboard round to guide knife), stack and frost. And/or use the odd shaped leftover pieces for one of the internal layers.
Possible pans - lasgana shape (9x13), sheet pan (cookie sheet with edges all around), or aluminum foil roaster pan (various large sizes).

**One more thing!
The recipe just says flour, so would you guys recommend using all purpose or cake flour? Or half and half?
Thank you! :)