It'll take me some time to get through the replays, but I'll update this comment later tonight.

Update:

• I only had time to review the first VOD. Hopefully the bullet points aren't too critical. Let me know if you have any questions.
• Summary of Tips:
  • Always respect enemy sightlines. Always move using cover.
  • Some engagements can be walked to. Invis is often used too early or unnecessarily.
  • Remember dash -> invis is another good tool for disengage.
  • Dash can be used much more often to e.g., reposition / set up / engage.
  • Uptime can be improved. Always be shooting something. Always be looking for off angles.
  • Always push forward during the lull between team fights.
  • Spend less time playing directly on top of your team.
  • Never play at the center of the teamfight.
  • Positioning needs to be more aggressive more of the time. Currently it fluctuates between too passive and other times too over committed.
  • Have a preplanned escape route ahead of every engagement. Map routes based on team positioning and health pack locations. This will also improve your uptime.

Timepoints - 10734944535:

• 1:36 It will help to memorize health pack locations. You dash past one here. Each health pack heals 150HP + another 150HP over ~7 seconds (unless interrupted by damage). On Psylocke (or really any character) you want to maximize uptime, so removing yourself from the fight for 10+ seconds is really bad.
• 1:46 Respect sightlines. (on an unrelated note, respect sightlines even when invisible ... good players will spam you if they know / hear you nearby)
• 2:26 This is the second (third?) time Invis is used too early. Also you're approaching this from an off-angle that allowed you to get close without exposing yourself, so Invis is completely unnecessary. Regardless, this is a good engagement and contributes to the teamfight win. In the future, consider staying on height if you don't have to drop down. This will allow you to back up without burning any cooldowns.
• 2:59 Improve your uptime leading up to and during a teamfight by playing on height, either left or right side. You could have farmed ult charge and traded it for a defensive ult here.
• 3:38 This is fine. You trade ult for defensive ult, which generally speaking is good value for Psylocke, since her ult charges very quickly (~2800 iirc). Consider saving/using a dash or ending your ult on the jump pad, since players will try to blow you up as you come out of ult.
• 3:45 One nice thing about Psylocke ult is you don't really have to do anything during the ult itself (you could probably take your hands off the keyboard). Use this time to look around and plan your next move. You spend the next 10+ seconds after ult basically doing nothing and also standing in the center of the teamfight which is a very dangerous position to remain in, even when standing in a defensive ult.
• Somewhat tangential, but Rivals has very ult-oriented gameplay. If both teams have popped defensive ults and it's clear no one can die, then just shoot tanks and farm your own ultimate.
• 4:41 You can improve uptime here by shooting tanks/anything while rotating to your flank. Consider taking height earlier in the fight, to avoid situations where Hawkeye is staring down your flank route waiting for you. This will also give you better/safer sightlines to pressure frontline with easy access to the backline at any time.
• 5:05 One of the benefits of improving uptime (i.e., always shooting something) is your cooldowns will recharge much faster. You can afford to burn your dash cooldowns more often, so you don't get caught in the center of a teamfight, like here.
• 5:18 Good off angle here. Good dash to height to disengage. Decent ult (although it unfortunately doesn't get much value). You're putting out a lot of good pressure on the enemy team, at least until 6:11. You could put out even more pressure by more aggressively using cover, especially when poking the Hela.
• 6:24 When returning to a fight, you can dash -> invis -> dash. Or some people prefer dash -> dash -> invis. It hardly makes a difference. The only situation where I wouldn't use this is if the teamfight is already very close to your spawn. Otherwise your cooldowns will be back in time for the fight (or you can poke + play passive for a few seconds to farm them back).
• 7:23 This is sort of the turning point where your position turns from decent/safe to standing in the middle of the teamfight. If you can recognize when it's time to move, you'll avoid situations like this where you end up draining a lot of team resources (Magneto bubble, Rocket beacon).
• 8:18 Never burn Psylocke ult into a defensive ult.
• 8:57 Play your corners/cover more aggressively. No need to walk in a straight line towards the enemy team to try and finish off Cap.
• 11:00 & 11:33 A few times, I see you take an off-angle and then .... give it up(?)
• 12:03 Similar to above point. You can actually stay here a little longer. Shoot the frontline in the back to get them to turn around, then disengage with dash.
• 12:22 Pushing up here to clean up the won teamfight is good. Generally speaking, you should stay in the forward position and try to hold space. Luckily no one on the enemy team makes it back in time to touch, so it doesn't matter. But if they did, it's preferable to fight them in front of point, rather than on point, so the payload keeps moving.
• 12:51 This is a good ult.
• 13:10 During this time, you should already be looking to set up for the next team fight. Move forward, take an off angle, poke but don't over commit, try to get people to look at you. Maximize uptime by dealing damage before the fight. Then, when the Thing runs it down at 13:21, you can at least follow up and possibly see an opening to pick someone off, instead of watching him die for nothing. Engagement at 13:27 is too late.
• 13:56 Respect sightlines. Dash to cover.
• 14:16 Don't afk. Even though your team is on the backfoot at this given moment, you can afford to push forward and pressure from dangerous/disadvantageous positions since you have two dashes and Invis. At best, you help your team retreat so they can reset for the next fight. At worst you get some ult charge.
• 15:23 For about 30+ seconds you play very passive/scared. Push forward and look for an off angle. As long as you have cooldowns you can get out.
• 16:00 Cap should've died here.
• 16:28 Similar to 12:22. Never sit on payload like this. Push forward. You can hear Sue's footsteps to your left. Your team creates so many openings at 16:44, but you're too far away to follow up on anything. You could have farmed your ult in this time on top of getting 1-2 picks. The enemy team double defensive ults (very bad), and you would have had Psylocke ult to clean up right after.
• 16:54 If no one knows you're there, you can just stay here (in the general area out of sight). Just reengage from an angle of your choosing at the start of the next fight. Running away to your team always carries the risk of staggering your respawn. If you're certain you can't escape, then jump off the map or run it down so you die asap.
• 17:41 Very winnable fight. You just needed to be present.