Do not book separately. Book all under one reservation. It makes things a bunch easier. If you are really that likely to lose tickets during the few days of your trip, then there are other issues that Amtrak can't fix.
There are rarely any benefits (excluding itineraries that are part AGR, part paid) to booking separately.
Arrow will calcluate fares based on an itinerary, and may give you a better deal than bookign separately.
When you have same-day connections, having everything on one itinerary puts your name on the connection report for a train (and "linking" reservations does not do this, only booking as one reservation will), so they know to expect you if there is a chance of misconnecting. Separate reservations doesn't invalidate any guaranteed connectons, but putting them on one makes it a hell of a lot easier.
Lastly, suppose you had separate reservations and didn't print your tickets, and one of your trains was about to misconnect and they had to bus passengers on a shortcut to make the connection. Well...now you don't have the ticket you'll need to take that connecting bus to the next train. You can't print the ticket on the train. *Maybe* you'd have time to run into the station at whatever connecting point they pick and print it there. Still, why bother with that hassle.
Just put the tickets in your wallet. How often do you lose your wallet?