Try to get Forest of Thorns done first for Rowanne, which gets you access to the Golden Cog/Clockwork Sphinx/Rowanne trio (last slot is discretionary), which should be able to get an easy win condition of pump/cast Rowanne for all quest fights (and a lot of other content it can be used for)
Before then, troops that are boosted off things such as board state or gems destroyed are probably your best bet, with some control. My early game setup for my last early game playthrough (granted, this was a long time ago) was Hero(priest’s hammer)/Lion Prince/Tyri/Dragonian Rogue. Pick a class with the War talent tree for +4 hero attack in all battles (warlord works) and put some common ingots (won in PvP, shouldn’t take too long to get the few that you need) on Priest’s Hammer. A priest’s hammer +5 is an additional +7 attack. This should give your hero a fairly hefty skull strike. You’ll want the Forest of Thorns banner, and Tyri leveled high enough to destroy at least 8 gems, so if you match green and get a surge, you have enough mana to start casting, otherwise you’ll need a few matches to get one of your other supports up. Once you do, you can usually ping between your three troops to deal damage while getting back minor amounts of mana and/or setting up skulls for the hero, covering all colors except one as minor generation/control, with four of them also leading into damage. Don’t bother casting the hammer or matching browns unless you have only bad moves left.
I also used an early game Dragotaur build in the past, but this was back when quests were super easy and you could unlock Dragonguard early. The build was yellow weapon hero + dragotaur + dragonian rogue + baby dragon (or some other red colored dragon, like Lady Garnetia). Without access to a Dragon Hero, you might still want to throw a hero up front because of the high team attack bonuses, but you don’t want to block brown, crude club as a common rarity club could work there. If you don’t have D. Rogue, you could try any Dragon’s Claw dragon you do have that covers a color you need, even if it is just Dragon Eggs for safety. A big problem with the dragon setup you are using is that you are mana blocking dragotaur on brown, taking away yellows that Dragotaur gets a big boost off with Drake Rider, and don’t get much out of the red either except a direct damage that probably won’t kill. Dragon setups can be highly effective early game because of dragotaur boosting attack and fishing skulls as your win condition, generally pretty slow but it works, you just want him not mana blocked but as far back as you can still be “safe”. Also Drake Rider isn’t from Dragon’s Claw. Three dragons from dragon’s claw on the team at early game is a not-insignificant boost that should help out, hero up front (crude club, cheap mace weapon for attack to not block brown, high starting attack even if you don’t have dragons), at least far enough to get Rowanne and get that team going. Match priority should be Skulls > Dragotaur Mana >>> everything else.