Quick background, I placed in low silver this season, ~1560, and have climbed from there to plat, almost entirely in solo-q. I have a general understanding of the key differences in the gameplay of CS, even though I haven't played much myself.
Climbing in Overwatch around those ranks relies on two things in my experience - hero mastery, and communication. Filling and telling your team to group up is a great start, but sadly is often not enough.
Hero Mastery: when I started the season, I had a pool of around 5 heroes that I played a lot in quick play and was comfortable on. I looked at guides on how to play those heroes better, watched streamers, all the usual stuff. I picked from my pool the hero that I felt best fit the situation, and made suggestions on what other picks would help the team comp. I almost never filled onto heroes that we 'needed' but that I wasn't comfortable on, unless it got desperate. Being confident on these heroes meant I could make individual plays and carry my team. Bear in mind that some heroes can carry more easily than others, but all heroes can carry when you are good at them.
Communication: this is very important, as most people I have found in lower gold and below have poor communication. Positivity is important, because tilted teams are much more likely to lose. When discussing team comp, try to avoid criticising specific players or telling them to switch. If your team needs a Rein for main tank, I often just ask if anyone is good on Rein and then try to work from there, and failing that will ask if one of our dps can switch to tank or heals. Maybe we get a Dva and have to work with that, but letting your teammates work to their own strengths is powerful. If you end up stuck with solo heals, sometimes you need to give it a shot, just so long as the solo healer is either Ana or Mercy for the higher hp/s. I don't know if I just got lucky, but I managed to avoid having 4 or more dps in pretty much all of my games since developing communication in this way. Some of this may seem basic, but it made a big difference in my games.
More on comms: make callouts, even if you're the only one making them and people aren't responding. People still sometimes follow through without responding. Call out low health, out of position or flanking enemies; suggest ult combos and positions i.e. high ground; and when a fight is lost and your team needs to regroup as 6. Saying the number 6 is significant, as some people consider 4 people going in with the tank at the back to be "grouping up". I can't really compare this to CS, but that's what advice I can give for getting strangers to cooperate in Overwatch.
Also be sure to use prefer player on good teammates, especially tank or healer mains, and don't hesitate to block throwers. Until the SR and queueing system gets some work, this is one of the best ways to improve the quality of your teammates. Good luck and I hope this wall of text had something useful in it!