As stevep has already stated, the crisis was bound to arise with or without Phocas. Peter, Maurice’s brother and the commander of the Danube armies at the time of the revolt, was an incompetent wreck, unable to defuse the situation (Priscus, who’d go on to become Phocas’ Comes Excubitorum, did quell a revolt by disobeying the exact same order to winter across the river in 593.) Now, once the rebels marched on Constantinople they publically proclaimed their willingness to have Theodosius (Maurice’s son) or Germanus (Theodosius’ father in law) be crowned in Maurice’s stead. But the Emperor mistrusting his son had him flogged, and Germanus choose to hide in Hagia Sophia. Then the situation spun out of control, and led to OTL events.
What could happen is that Maurice gives the throne up willingly to Theodosius. Problem here is, that Theodosius would be in a tough spot, and would have to make large concessions to the undisciplined troops. Same would apply to Germanus. This could certainly relieve some pressure on the Avars, as the campaign would be called off, and not continued until April-May 603; if that. Whether or not the offensive is renewed the following year, the Avars would be badly hurt, and unable to cause any trouble for a while, unless the Balkan armies are rotated to the East.
On the eastern front, Khosrau might grow bold and decide to strike during the turmoil. His casus belli would have much less legitimacy in this case, and Roman resistance might stiffen (a really messed up scenario could have Maurice calling on Khosrau to overthrow Theodosius…) Should Maurice go peacefully into retirement, the Empire would not undergo a civil war, such as it did IOTL between Phocas and Heraclius; as a result, the Great War might just become another of the many on and off engagements between the two powers, and by default leaving them in a stronger position by the time the Arabs, if Islam still develops, push north.