I don't believe that your existing save files have an effect on other routes. And you do make very few decisions throughout these stories, the lack of control seems to bea big part of Echo.
The routes themselves exist in isolation from each other, so doing the other routes won't give spoilers to Jenna's route (i.e. the situations faced during the TJ route are very different to the ones faced in Carl's route, so you won't get any tj spoilers by doing carls route)
The route you do is determined by your response after Flynn causes a scene and storms off, and some routes have multiple endings while others do not (listed below). You do not need to start a new game if you make a save at this choice since the introductory section remains the same no matter what and the storylines only branch from there, so you can just load that 'choice save' and start a different story. (you may need to make 1 new game to get back to this choice however)
TJ: find TJ (1 ending)
Flynn: Go after Dlynn(1 ending)
Carl: Sit with Carl (multiple endings)
Leo: Follow Jenna and Leo > Go to Leo (multiple endings)
Jenna: Follow Jenna and Leo > Stay with Jenna (no endings yet)