Unless the rules have changed, I believe the TD was correct in their answer to you. In USSSA, players are free to play with other teams as long as their home team is not in the same tourney. If the team they are playing for would win a berth, they need to declare if they plan to continue with that team and represent them at the national level or their name will be dropped from the roster making them ineligible for national play with that team. If they decide yes, they need to get a release from their home team and the new team needs to file that request and do the paperwork to get the kid signed up on the proper roster for participation the rest of the year. To my knowledge, all the USSSA rosters are online for coaches and administrators to see. It's a really cool system with teams earning points for weekend competition and accumulating points depending on level of play, how you are registered, and ending the year with a points champion as well as State and National tourney champions.
I believe all USSSA teams can enter the State tourney if done so by a cut-off date. This is the part I'm unsure of; I believe only the top finishing teams at the State have their rosters frozen and if they decide to continue to the national tourney the kid/kids in question would be locked into that roster. If the teams turn down the bid, the pick up players are still free to return to their original team if they choose and the team that turned down the bid is no longer eligible for post season (national) play that year. USSSA used to and I assume still host tourneys that can qualify you for a trip to their version of the World Series. It is always a carrot dangled at the State tourney. I believe the State tourney champion, perhaps 2nd and 3rd are also awarded a bid. Perhaps the teams in question do not have the finances to attend so they turn down their opportunity, those slots can either be awarded to the next best teams that finished in the State or be awarded to the winners of other sanctioned USSSA tourneys depending on the format decided on by the State Director. Also, I believe the State Director has the power to assign "at large" bids to a couple of teams.
Remember, this is old school USSSA as I remember it and things might be different. The points thing is something I thought ASA and others would get involved in. I really like it as well as the roster situation that allows some freedom for the kids to have fun throughout the course of the summer. The national tourney is a huge event. Depending on age group, there could be upwards of 125 or more teams playing for the brass ring. I liked the format and thought it was great to promote the sport.