The Heavensward Leve's all have HQ main and off hand in their reward rolls. So you will find them on the MB for next to no Gil. It's pretty common to get them on triple turn in as well, so people end up with 3 just off a single Leve.
The electrum accessories and HQ patrician combined with purchased, leveled main and off hand will let you HQ the higher sets yourself.

I'd still suggest pushing forward with the MSQ until you unlock HW or even Storm blood. level 21 to 60 in Firmament took me 2 hrs. Granted it only took 4 I think doing it with ARR content but I made level 40 Firmament stuff with vendor gear to get to 60. You can do it on a base server as well. You will just need to make more of the level 40 items.