if the Ukranian armed forces are really intensively shelling the enclaves,

Looking at the OSCEs maps and comparisions to the sort of base line seize fire violations, I would rather say that we can know that the Ukranian armed forces has escalated the shelling. How intense it is, is hard to grasp.

and if this is the reason why the population are massively fleeing, spontaneously or not (if indeed they are)

I would say that it is more a question of evacuation, as I understand that the rebel authorities has declared that military age men will be conscripted and the rest evacuated. How much people are following those orders is unclear. The reasoning here on part of the rebels is clear, if they think an attack is coming, better to evacuate and mobilise ahead of schedule.

then why are they doing so?

This is where I fear that the conflict takes on a life of its own and that the Azov Battalion or any other sufficiently motivated group can start the war simply by increasing the seize fire violations. Its not like the rebels won't return fire, indeed they are primed to do so. And any Ukrainian military unit coming under attack is primed to shoot back. And militaries has plans for different scenarios and if they think something is happening, they will execute those plans.

And that is in the best case scenario, in the worst case scenario the Ukraian military is intentionally attacking (but it is starting somewhat cautiosly for some reason). I don't know why they would do that, but militaries tends to have among their plans include plans that are not always very rational on the larger scale.

Defence Scheme No. 1 has a place of honor among crazy military schemes:

Defence Scheme No. 1 was created on April 12, 1921, and detailed a surprise invasion of the northern United States as soon as possible after evidence was received of a US invasion of Canada. It assumed that the US would first attempt to capture Montreal and Ottawa and then Hamilton, Toronto, the Prairie Provinces, and Vancouver and the rest of Southwestern British Columbia. Defence Scheme No. 1's US counterpart was War Plan Red, a plan to invade Canada as part of a war with Britain that was created in 1930.[1]

The purpose of invading the US was to allow time for Canada to prepare its war effort and to receive aid from Britain. According to the plan, Canadian flying columns stationed in Pacific Command in western Canada would immediately be sent to seize Seattle, Spokane, and Portland. Troops stationed in Prairie Command would attack Fargo and Great Falls, then advance towards Minneapolis. Troops from Quebec would be sent to seize Albany in a surprise counterattack while troops from the Maritime Provinces would invade Maine. When resistance grew, the Canadian soldiers would retreat to their own borders, destroying bridges and railways to delay US military pursuit.[1][2]