Does SR from one role affect another?

Winning or losing is the majority factor in changing your MMR, which in turn does the same for your SR.

When you started placing your SR (and MMR) will swing a lot so that’s not a surprise, it will level out over time. Your MMR is split for every single game mode Open Queue, Comp Role Queue(Support, DPS, Tank), Arcade, Comp Arcade (also split between 3v3, 6v6, CTF, Lucioball, ect), and Quick Play(DPS, Support, Tank).

Each of the values start you at the equivalent of 2350sr and move up or down from there as you play. PSBR(personal skill-based rating) only effects a small minority of your overall SR gain or loss, roughly 5-8sr.

Your main focus should be learning the game, staying alive/knowing when to fall back, practicing your aim, and supporting your team - anything that leads to a victory.

For more information:

https://us.forums.blizzard.com/en/overwatch/t/how-competitive-matchmaking-and-rating-works-season-26/