I do think about Doggerland, fairly frequently! And I feel it’s important to point out that its disappearance absolutely was because of “the slow heating up of the Earth leading to a gradual melting of ice and snow causing the rise of the ocean”. By the time of the tsunami, the land area would already have been reduced to a handful of small islands.
In general, a true tsunami (caused by displaced water) is never going to permanently submerge an area of land, because the water displacement is temporary and it will return to where it came from (causing huge damage on the way, of course). However, with islands that were already barely above sea-level, it’s definitely possible that such a huge inundation just obliterated their topsoil and left them below water level.
It’s hard to say, though, because the end of the last Ice Age was a wild time for water moving around and getting into places it shouldn’t. In particular, as the glaciers melted you’d often get huge meltwater lakes forming behind dams of ice and rubble which might then burst, suddenly dumping nearly inconceivable amounts of water all over the downstream terrain. One of these bodies of water, Lake Agassiz in North America, was so huge that it covered multiple states, and it’s thought that it drained very suddenly into the Atlantic ocean about 8000 years ago (i.e. contemporary with the flooding of Doggerland), dumping so much water into the ocean that global sea levels rose 1-2m over the course of a couple of years.
And if you’re wondering - yes, it’s theorised that this event, or the combination of the others like it, was responsible for the flood myths that exist in so many cultures. Coastal and lowland settlements around the world would have found themselves in a period of time where the sea just kept rising - not overnight, but inexorably and seemingly without any end in sight. Many, many cultures would have retreated to higher ground, only to find a few months later that the water was once more lapping at their doors. In some ways I find that even more profoundly terrifying than the idea of a wall of water sweeping everything away.
