A terrible, terrible catastrophe.
The Common Army wasn't THAT badly equiped. They had enough modern rifles for all of their soldiers when the war started and, when equipment attrition massively hit the army after the disasters of 1914, like it did every armies, it recovered in a decent time and even then, I think they still had enough old rifles to arm the rear troops until production caught up. After all, Steyr and FÉG produced an enormous amount of M1895s. Meanwhile, Russia is notorious for not having enough rifles for their massive army in both world wars.
Its problems equipment-wise were in modern, quick-firing field artillery and, later, uniforms. Škoda was doing its best, but the governments were all too frugal when it came to modernising the Artillery and they were short of guns nevertheless despite having good designs ready. Still, they had top-notch Mountain artillery and Siege Artillery.
Uniforms became a problem later in the war as they switched from the dark-blue uniforms to field gray/green. A-H industrial capacity struggled throughout the war, but that has little to do with the frugality of the governments over the Common Army and more about industrial limits in a rapidly-growing economy. Honestly, I think the equipment issues are overstated and little different from the other armies in the war, just like the linguistic issues (although the A-H commanders really fuckes up when they blamed Czech soldiers after the initial battles with Serbia).
Anyways, having no Common Army is a catastrophe waiting to happen. The survival of that army until 1918 is what kept the Empire together for so long despite everything. Plus, having two separate armies sounds like a logistical and leadership nightmare. Without a Common Army, is there even a common General Staff on top of all branches? If so, maybe I'm being a bit too critical and pessimistic, but it would still be a problem supplying units that are in completely different insitution. At least, with the Common Army, there's a cadre and Staff ready to command all the branches into cohesion. I really don't think the Landwehr and Honvéd can have much cohesion without the Common Army and its Staff, which is a disaster waiting to happen... Even if their is a joint staff, I can imagine it having serious trouble getting both branches to cooperate: Getting Honvéd units to be transfered to the Isonzo front of Galicia or Landwehr units in Vojvodina or Transylvania...
Unless the two branches and a joint staff can create some kind of unity between them all, then I would stand corrected, but I sincerely doubt it even if it's hard to imagine Austrian and Hungarian forces in a worse situation than after the inital OTL battles.
Honestly, WW2 Germany is the perfect example of how wrong things get when there's so much competition between branches.
However, there is absoluty no guarantee that A-H becomes a puppet of the Germans. If the A-H forces perform better and suffer less catastrophic casualties in 1914, then it simply won't happen.
Edit: I forgot to mention: I remember reading that Hungarian officers were often looked-down upon by other Magyars for serving in the Common Army rather than the Honvéd, with the latter being more honourable and 'Hungarian', explaining why the Hungarians were under-represented in the Common Army. Making the Common Army more attractive for Magyar officers would have worked wonders.