Was the military situation in December 1776 really as dire as American history books have made it seem, or is it just propaganda to some degree?
I would say for the most part, yes, it was that dire. Washington's army by early December of 1776 was hemorrhaging men and supplies at a rate that could not be sustained. His army of nearly 20,000 men had been beaten continually from pretty much the spring of 1776 straight until Christmas 1776, and by that point had only a couple of thousand soldiers still fit for duty. Many had run off, been killed or captured, and British General Howe was on the other side of the Delaware River with 20,000 well-fed, well-supplied, well-rested soldiers to chase Washington with.
What's more, the enlistments were up on December 31st, and those not already done with their service could simply hang out a few more days and go home with their commitment honored. To top it off, Washington had lost a couple of his better commanders to British capture or surrender in the weeks before the Battle for Trenton, and he was down to 18 cannons.
All of that being said, I think American History books do lean a little towards the dramatic if not outright propaganda at times, overlooking Washington's flaws and mistakes in favor of having a national founder and hero.