'badly disturbed' is correct and can be used in written English.
It is usually used to describe a person's emotional or mental state, or to describe a disruption to something. For example: Her sleep was badly disturbed by the loud noises coming from outside.
Badly disturbed hillsides and abandoned paddy fields.
The Dinaledi chamber has been visited by explorers in the past, and the soft sediments in which the bones were found have been badly disturbed.
Recent ground-penetrating radar survey of selected mounds at Effigy Mounds National Monument reveal that many are badly disturbed, but others appear to be comparatively intact.
Although the deposits within the broch had been badly disturbed by earlier attempts, this work revealed a complex sequence of construction from different periods.
The changes have badly disturbed the ecosystem in Grand Portage, resulting in an invasion of gray squirrels from the south, the total depopulation of trout in some area lakes, and an exponential increase in deer.
Also, sleep was badly disturbed.
We realize that your mother's recent comment that your cousin Rachel is "aging badly" — despite the fact that she's a college sophomore — was seriously disturbing.
But, because these sites were long known to local people, they had invariably been disturbed, if not badly looted.
He or she may have behaved badly when their state of mind and emotions were disturbed and in turmoil.
"The hides on the outer edges of the rolls, as well as others that were disturbed as the vessel sank, had all deteriorated badly," Mr. Brown said.
Amid the chaos, a wasp nest in a nearby tree was disturbed, and a number of Australians were so badly stung that they were incapacitated and also needed to be evacuated.
