No, the distance being contracted doesn't mean you only have to cross 2 light years which would take just over 2 years at your velocity, either that statement is not true, or I am quite mistaken about special relativity.
If you crossed 640 light years fast enough that it was contracted down to 2 light years in your frame, while avoiding any time dilation strangely, and you then crossed THAT distance fast enough that you took just a little over 2 years to do so, wouldn't you then experience time dilation on THAT duration, and observe quite a bit less than 2 years during your trip?
If the 2 light years in your frame was 640 light years for a signal laser you fired on the same trajectory when you left, then your trip would take more than the 640 years required for a detector at your destination to receive the signal.
Yes, in your frame, you'd only experience/age/observe 2~ years, and claim your laser beam only crossed 2 light years.
If you then turned around and fired a reply laser while going fast enough that you again measure the beam traveling for only 2 years as it crossed 2 light years, you would return 1280+ years after you left, roughly 4 years older.
If the contraction observed from your frame meant you only took 2 years to cross that distance without any effects from time dilation, you'd arrive at Betelgeuse 638~ years before the signal laser, and you'd make it home a thousand years or so before you even left!