I believe that 1246 is needle lift sensor circuit range/performance it's aka VW 17654 code. The glow plug light will show a bunch of faults but the glow plugs never are the cause of bad running after the engine is warmed up. My guess as to why they are wet are it's fuel or oil.
About the light, my guess is a wiring problem. Clean the battery terminals, the glow plug harness connectors, and the plug that is attached to a wire coming from injector #3. Read the injector/nozzle FAQ in the FAQ list at the top of the forum and it'll explain how it works and might give you a clue as to why this fault is being thrown.
A broken or frayed wire could cause this fault. Believe it or not, a bad taillight can also cause the glow plug light to flash. A more serious problem is a bad injection pump or bad nozzles. The nozzles can be cleaned and at your mileage it's a good idea to replace them if they haven't been replaced yet.
If you intend to work on your car any more than checking for codes getting a vag com will help tremendously. An obd2 scan is a good start but you can't check engine timing or other things without a vag com, see here for vag com.
http://www.ross-tech.com/ Yiou don't need the expensive cable, the one for $250 is fine. Here's a list of stuff that it does on the TDI
http://www.ross-tech.com/vag-com/cars/tdi.html
The hesitation might be a boost leak. Check for loose or broken hoses along the intake tract. Also check all the vacuum lines for frayed ends or cracks or rub throughs. Your car is now 10 years old, give it regular maintenance and it'll take you another 10 years!
