Well, we got to the point where they're doing my dishwasher and we find that it is more than 10' from the dishwasher if we run the drain line along the back of the cabinets in an L-shape to the disposal (more like 13'). The installation instructions say no more than 10' and when we called GE to ask, they said that was a hard limit. So, first question to all of you experienced in plumbing: is that 10' limit really hard and fast or will I have trouble by extending it to 13'?

The only other option we've come up with is letting it drain below into the basement where the laundry machine drains upward. The laundry machine's drain is almost directly below.

Is there anything else I'm not considering and should?

I don't believe that there is a rule that says you have to drain into the disposer.

Dish washers require an indirect waste and may not be connected directly to the DWV system.

At least a high loop and GD installation, or the complete air gap method.

you can run you hose around in less than 10', by cutting the corner short.
Holding it up against the underside of the counter will keep it out of the way if this corner cabinet has stuff in it. This maintains your high loop for a longer distance, which is fine.

About this: "...At least a high loop and GD installation..." It's wrong.
Millions of dishwashers have their drain hose connected to the drain plumbing under the sink strainer, not through the garbage disposer.
What is right is that dishwashers "require an indirect waste and may not be connected directly to the DWV system."
To see images of the fitting, search using key words "dishwasher Wye" or "dishwasher Tee" . That is all you need. Any connection prior to the P trap is by definition an INdirect Drain. A GD is not required. (b.t.w. this may help you gain a little bit more room.)

The next option is if you still haven't got enough distance. The P trap can be moved. It does not have to be located directly underneath the kitchen sink drain. Plumbers put it there because it's then easier to see into it and snake into it. It's not because of any requirement.

Another option is to use radiator hose instead of the corrugated hose that comes with the DW. It'll drain better.

And they are almost always required to have an air gap in the counter.

So, no, it is wrong to say "they are almost always required to have an air gap in the counter."

In places where an air gap is not a required element, dishwashers have their drain hose connected to the drain plumbing under the sink strainer, and not through the garbage disposer.

So it is wrong to specify "GD plus high loop".
It is also wrong to imply that an air gap is required when there's no GD.
It is also wrong to say that most are required...
Where a DW is not required to have an air gap, you don't need to have a GD either, to connect the DW.

It is not allowed by most codes.