Why do our clocks not go out of sync with the day over a year?
You are correct, and in fact the clocks are offset. But most people never notice this because of the way days are defined.

If you measure the "day" length by the stars (instead of the sun) you will see it's not 24 hours at all. It's around 23 hours 56 minutes. (Check this by going out at night, picking a bright star, and noting the exact time it passes due south. Then observe it the following night and note it passes 4 minutes earlier.)  By the way this shortened "day" is the sidereal day.

Each day, the Earth shifts along in its orbit (by about a degree, as you correctly observed in the question). So if you go out and point at the Sun at Noon every day, you are pointing in a slightly different direction. (I'll also neglect the Solar Analemma at this point too as it's just too confusing.)

Really, the "day" of 24 hours is not related to the true rotation period of Earth at all; loosely speaking it's just the average time between sunrises, or sunsets, or noons.