Call 1-800-USA-RAIL to sort out the pricing/availability issues. The price is likely going up each time you check because those are busy dates and they're booking up fast. Also ask about the baggage policy for sleeper passengers. I think you get some more allowances with a sleeper since you have more room to stow the luggage.

As for hauling the 'stuff', from the moment you get out of your car, a redcap will haul whatever bags you want, to wherever you want for a nice tip. In your case, that's probably worth it. You can also arrange for a pickup at your destinations and never have to lift your bags until you get to your hotel/destination.

But as others have said, you may be better off enduring a few hours of airport headaches than a few days of riding the train with an impatient 2 year old.....