You aren't alone.
It can get still more frustrating, so take a deep breath and decide what you're willing to do. Most districts have limited classroom space to use for K (amaller tables and chairs, separate bathrooms, access to a separate play area, etc), so with half day they gain the ability to have slightly smaller class sizes and nearly double their capacity. Besides, do you really want to force your child into an already crowded all-day-K classroom?
You are probably too late get a variance to another school district that might offer an all day K program. Even if you are in time, that means you have to provide all your own transportation to the other district and you'll go through lots of gas $$ since you probably can't carpool with neighbors. Also consider that your child will have to get up that much earlier in the morning to do the commute to another district. How will this commuting time-hit affect what time you get to your job as a nanny?
So say you decide you can live with the half-day thing where you'll probably have guaranteed bus service unless you live within a half mile of the school. That's not the end of the frustration story. You get your son settled into half-day, then you find that the schools sometimes switch things around and make your morning child attend in the afternoons(or vice versa) to average out from school conference times, late starts, and early dismissals. Or certain days they have no school at all because it's an "afternoon only" day.
And it's a good thing you watch your own boys, because if you need your child to be in daycare to allow you to work full-time, then some private daycare centers require you to pay the same price for full and half day (because they have to hold the spot for your child and/or possibly charge you for transport to/from school). And if you think your kid can get into a less expensive afterschool YMCA daycare program, well, you would have needed to get on that list back in Feb. when you signed up for school. And because the school calendar has more days off than most women get in vacation each year, working moms have it especially hard.
Hang in there and gather your patience. Many PTAs offer after-school enrichment programs such as for sports, music, and art (even for K) that are no or low cost, and that might be helpful to your son's growth.
-B.