As already stated, the post-ARR story will get streamlined because many players are frustrated with it.
The ~100 quests take an extremely long time and with the frequent trials and dungeons you need to do, they take even longer for a dps because of the queue times.

However, you can try to use the Party Finder (not Dungeon Finder, the one where you queue and get paired with randoms. I think you confused those two) to get help.
While I didn't need to during the MSQ as I was playing as a healer, I often used it during my ARR relic where I had to run all the 50s dungeons and found help really quickly. A lot of max level players regulary browse Party Finder for various reasons and many readily help with a quick unsync run to get you through a story dungeon.

I would also stick to MSQ trials/ dungeons for now and not try to get into all the 50s dungeons you unlock after completing ARR. Escecially old max. level dungeons often have nightmarish queue times because they only pop up in 50/60/70 roulette which isn't done as often as other roulettes.

I also recommend finding a FC. It makes sloggish times much more bearable if you are surrounded by nice players and can chat away while doing the 8348093485. fedex quest. They are usually willing to quickly get you through trials/ dungeons aswell. It's a matter of minutes at 80 while saving you tones of time.

Also, since Twintania is an EU server try to aim for EU evening times to get your MSQ trials/ dungeons done.
If your squadron is high enough, you can try to use that for the neccessary dungeons, though not all are accessible via Squadron and you will still need to queue for most of the MSQ dungeons. Traisl aren't accessible at all with a Squadron.
Another solution, though one that also requires time, would be to level a healer or tank alongside your dps as they have really short queue times with healers getting almost always instant invites even at night. Unfortunately players always gravitate towards dps classes so they are over-represented and have the longest queue times out of all roles.

At 71+ you can use trusts for dungeons and a few trials aside, you won't have an issue with queue times, no matter your role.