I was a DRG main in ARR and HW, but abandoned it in SB because a couple things about the job just felt bad.

The main thing that gets me is constantly losing stacks off the Dragon Gage every time Blood of the Dragon falls off during the long run between pulls or when someone in the party lags too far behind and the group is stuck waiting. Boss jumps are also a small concern. Every time I lose BotD for things like this, the disappointment just sucks the fun out of the job for me.
Also a small issue discussed in this thread HERE.

BLMs have had a good QoL for a while now. They've been able to keep Enochian up and build Polyglot stacks full time by using Transpose and if they accidentally let Enochian fall of, they still get to keep the Polyglot stack they accumulated.
Now in ShB, it is even better for BLMs. Using Umbral Soul, BLMs can begin every encounted with full Umbral Ice 3 and 3 Umbral Hearts and probably built up a Polyglot stack to two as well during the jog.

MNKs also are riding high after the last update. After getting their Greased Lightning stacks maxed out during a fight, they can just spam Form Shift while running from one battle to the next without ever dropping their Greased Lightning. I now run entire dungeons without GL falling off even once. The Meditation buff is also a nice time saver, making it easy to just tap Mediation once and spam Form Shift and be ready at 100% for the next fight.

TL;DR
My suggestions are:
1) Maintain any Dragon Gage stacks that are accumulated if BotD expires. (BLMs already have this) and/or
2) Lower Blood of the Dragon skill recast time to 15 seconds, a time chosen as BotD low end duration is 20 seconds. (BLMs can maintain using Transpose or Umbral Soul and MNKs now have a similar benefit via Form Shift)