Lots of good replies here, but I'll add my two cents' worth.
No surprise, but a roomette would definitely be preferable if you can possibly afford one. It would be worth changing your dates a day or two to get a lower price, if that is a possibility. Keep in mind, though, a roomette is a very small compartment, but you can spend some time in the lounge during the day. The advantages are the privacy, the BEDS, and also the included meals for all of you. The meals offset part of the cost.
Most people would find one overnight in coach doable with a ten-year-old, but I agree the baby complicates the mix. I don't know how strictly you're trying to stay on budget, but with a baby in arms I would caution against trying to bring too much "stuff" with you. I have three children, adults now, but even with the older child to help, managing to carry a baby and all those needed items, your purse, plus luggage is a lot. And you'll want to bring something to help occupy the ten-year-old's time when the train fascination wears off. (Time passes slowly for kids.) I'd suggest just planning to eat meals in the dining car or cafe if at all possible. You could still bring a few snacks for the kids. I guess I'm just trying to say adding food for meals and a cooler to all that would be more than I'd want to wrangle, not to mention the hassle of eating at your seats. If both children are being little angels, it might not be a huge problem, but they aren't always little angels, LOL. I know the ten-year-old can carry some things, though, but "less is more" sometimes. As a mom you certainly already know all of this, but just don't overlook it in the excitement of planning the trip.
I would definitely let Amtrak know that you will have the baby with you even if an extra ticket isn't needed. My husband had significant mobility limitations and the Amtrak staff were usually very considerate of that. On Amtrak, though, it's unfortunately hit or miss so YMMV for sure.
One more thing--I'm not temperature sensitive, but on my last Amtrak trip, the car was freezing. I took my nightgown from my bag and wrapped up in it. I wasn't the only one pulling out clothing, so keep that in mind.
I think overall this will be a great trip and will certainly be one of those "Remember that time we...!" things twenty years from now. You seem to have the right laid-back attitude, so go with the flow and enjoy. Just have reasonable expectations and make sure your older child is prepared for a l-o-o-n-g train ride.