Exactly right - think about what the visible moon actually is: daytime! When the moon is illuminated, you're seeing the daylight side of the moon, which means the exposure of the moon, to see all its details, needs to be the same as you would take at the beach on a sunny day.
The problem you're likely having is you are taking cameras in a wide metering mode, pointing them into the sky with a short lens, where the moon probably takes up less than 10% of your entire composition, and asking the camera to meter what it sees. Mostly, the camera sees a bunch of black sky, so it tries to meter for that...but the moon's surface is bright daylight - so when the camera meters the sky and takes the exposure, you end up with a slow shutter speed, and a badly blown out moon, probably with some motion blur too.
The keys to getting good moon detail are:
#1 - and the most important by far - the longest lens you can use! The 300mm lens would be your best bet. It you can get a longer one, do. The 18-55mm on the NEX is DEFINITELY not enough. The 135mm is still pretty short. You want to fill as much of your frame with moon in the viewfinder or on the LCD as you can...so the more lens reach, the better.
#2 - meter off the moon, not the sky! This can be done by the camera, by switching the metering mode to 'spot meter'...that will tell the camera to only meter off the very small center point and nothing else, so when you point that center point at the moon, the camera will determine its exposure only on what's directly under the little center spot. Or you can set the exposure manually as Carlos listed - keep thinking 'daytime' - lowest ISO you can, smallish aperture of F8 to F11, and shutter speed of at least 1/150 to 1/300 or so. If handheld, use the stabilization if you've got it...otherwise, consider putting the camera on a tripod, and using the 10 second timer to take the shot - that way, you can take your hands off the camera and give it time to settle down and not vibrate. VR should be off if using tripod, and best if you use the mirror lock up function too.
The longer the lens you can use, the more you can fill your screen with big, glorious moon. On Saturday, I used a 500mm lens with a 2x teleconverter for a total of 1,000mm to get this:
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses