"balls to the wall" means that you've run out of options, and there's no way out.

I think it instead means to go all out. Here's a little research paragraph from Slate Magazine (can I quote sources just like that??? I'm a newbie)

Somewhat disappointingly, it has nothing to do with hammers, nails, and a particularly gruesome way of treating an enemy. The expression comes from the world of military aviation. In many planes, control sticks are topped with a ball-shaped grip. One such control is the throttle-to get maximum power you push it all the way forward, to the front of the cockpit, or firewall(so-called because it prevents an engine fire from reaching the rest of the plane). Another control is the joystick-pushing it forward sends a plane into a dive. So, literally pushing the balls to the (fire)wall would put a plane into a maximum-speed dive, and figuratively going balls to the wall is doing something all-out, with maximum effort. The phrase is essentially the aeronautical equivalent of the automotive "pedal to the metal

The expression is first found in military-aviation sources that date from the Vietnam War, and it was recorded in the slang of U.S. Air Force Academy cadets in 1969. Although no evidence from the period has come to light, Korean War veterans have also reliably claimed to have used the expression in the 1950s. An earlier parallel is balls-out, in the same sense, which is found in military-aviation sources that date from World War II. (The phrase was also painted on the nose of at least one fighter plane.) In both cases it's likely that the possibility of an anatomical interpretation has helped the expressions gain wider use.

There is a better, much older answer. Though I can't give a reference, I have read it:

In the old days, when steam engines were being developed to generate electric power, way before air planes or the Vietnam war, they were housed inside buildings. Steam engines had two spinning balls on top that were used to regulate the steam flow and thus keep the speed constant. Those balls would fly in or out according to how fast it was going. When it was running at top speed, it was said to be "Balls to the wall", or "Balls out". It is now lewd or sexual, nor does it have anything to do with aviation.