firstly, dwarwes now have at least 2 mounted mercenary. you got the regular dale cavalry and you have another one in the lands of rhun (dont know the name or region because i screwed up my tatw 3.1 and have to reinstall everything).
in my dwarven campaign my tactic was to take every soldier that hasnt free upkeep and take the 2 rebel settlements in the west (nenuial and one more) and "wormcove" on the eastern part of the map. then i continued with those troops and took "dains halls" with the eastern army and "Carn Dûm" with the western.
!!!remember, the dwarves can easily win 1vs5 fights versus early OoG because they dont have those cursed snaga skirmishers!!!
after i have taken "dains halls" and "carn dum" i build up my western army with all the units i can within 2-3turns, (usually 2dwarven warriors,2 miners,1 axe-thrower) and with the help of only these "reinforcements" you should be able to kill off OoG. after that its just the OotMM left. against them i try to fight as the defender and almost always in a settlement, always have at least 1 unit of cavalry in the army to take care of those cursed "snaga skirmishers". i killed of the OoG within 35 turns and then slowly made my way down to moria. at turn 83 i could recruit my first vault vardens(didnt build barracks in erebor cause i felt like i needed more income, 5k + every turn wanst enough

) and soon thereafter i decided to try my luck against the balrog. so i put 2 vault wardens , 2 iron guards, 3 axemen, 3 axe-throwers, 1 crossbow unit and some miners+ warriors to fill the stack and besieged moria. (also had a second stack as reserves, mostly miners and warriors, so i could hold the settlement.) at turn 98 i launched the attack (with some help of my best friends the silvan elves)after sieging moria for 2 turns. silvans elves killed of the orc-scum while i killed of the balrog using my 3 units of axemen, 3 axe-throwers and the crossbowmen.
until i screwed up the whole game (at turn 126 or something like that) rhun only attacked me once but because of a well placed tower i saw them coming and quickly build up my garrison. after a crushing defeat they gave up and just send assasinations to kill my generals (they actually killed 4/5) untill i only had gimli left. after that i accepted my first adoption just to be sure of survival.(remember generals cost a lot of money, i only accepted 1 adoption)
also I NEVER EVER use autoresolve, it makes the game way to easy!
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so to make it a little shorter:
1. use your starting troops wisely and take as many settlements with them as possible.
2. dont recruit to many generals (dwarves live very long!).
3. use the merc-cavalry units against skirmishers.
4. dont attack rhun outside settlements (just leave them alone for starters and retake the old dwarven settlements in the mountains).
5. DONT USE AUTORESOLVE!!!
Good luck!