I just tried a Tucson to New York booking in Amtrak.com and it offers 2 routes...Train #2 (Sunset Ltd) to New Orleans, then #20 (Crescent) to New York after spending a night in a hotel in New Orleans. Same day connection not possible. The other routing, Train #422 (Texas Eagle) connecting to #48 (Lakeshore Ltd) to New York has same day connection in Chicago...up to 7 hour layover to do some sight seeing, subway/elevated riding, or simply strolling around downtown and enjoying some great Chicago food. So it basically comes down to a 'short night' in New Orleans (9:40 PM arrival, 7:00 AM departure) (or spend an extra night and enjoy New Orleans), or same day connection in Chicago. Either way, the Amtrak computer will put you in the correct car so you don't have to be woken up to switch cars or rooms at San Antonio in the wee hours of the night.
As far as your cousin going to Boston and you to New York, for the Chicago route, I'd suggest you both go to New York, then he takes an Acela or regional train to Boston if arrival is on a weekday (11:00 PM arrival or 12:26 AM). Why shouldn't he enjoy one of the most scenic rides in the East...the Hudson River. If you arrive at Albany on the weekend, all that's needed is a separate ticket Albany to Boston, which, on Sunday, will be a bus during August due to CSX tie replacement work on the route (Sun through Thurs). I don't think there'd be any pricing advantage of 2 people in the sleeper CHI-NYP vs 1 person + Open Sleeper to ALB. Or, if there is, it's quite minimal. On Friday and Saturday trains (#448 from ALB), it will be coach or business class seating. BC is roomier seating plus free non-alcoholic beverages in the lounge car, which 1/2 of which is the business class seats.
Going via New Orleans, the Crescent has a 1:46 PM scheduled arrival at New York Penn Station. There's numerous trains to Boston from there that arrive at Boston at a reasonable time. The trip along the coast line, especially beyond New Haven, has numerous views of the Atlantic Ocean and its many inlets, tidal flats, and seaside towns.
As for booking it through Amtrak.com, except for an 'open sleeper' ticket, it should be easy to book as two separate reservations: TUS-NYP for 2 and a separate one for ALB-BOS or NYP-BOS for your cousin. If you decide to go the open sleeper route, you'll have to call the 1-800 number to make the reservation.
One last thing...AMSNAG. AMSNAG allows you to look at different options over a 30 day period. This close to departure, the prices have already started 'going up' (supply and demand, just like the airlines) and vary from $1300 to over $2000, depending on which date you want to travel. Being able to see in advance and compare makes choosing the best/lowest cost date that fits your schedule a breeze. You'll still have to book the trip on Amtrak.com, though. Click on the link that follows and enter: TUS NYP 08/01/19 30 then choose the #2/20 or the #422/48 options to see the pricing choices. It also offers other round-about routings as well.
http://biketrain.x10.mx/amsnag2.0/amSnag.php
Enjoy your trip!