The best way to survive on an otherwise hostile planet is to build a space base with an enclosed ecosystem there. The best way to keep a planet's atmosphere from escaping into space is to enclose parts of the planet, or the entire planet, with a giant roof or roofs to keep the air in, and to keep repairing the roof(s).
But if you are going for a more natural way to have a partially habitable planet, I should point out that Earth is mostly uninhabitable for humans, though almost entirely habitable for various lifeforms. The oceans produce a lot of oxygen we breath, and water evaporated from the oceans is the source of the fresh water we drink to stay alive.
But if you are teleported to a random position on Earth, it will probably be a spot on the surface of the ocean, and you will probably drown. An d if you are teleported to a random position on land, there will be a significant chance you will die of thirst, starvation, or exposure to hostile temperatures.
I would suggest that for your planet, it might be good to calculate a mass and volume, and thus density, that gives it a somewhat higher surface gravity, so that the atmosphere is more compressed, and also a somewhat lower escape velocity so it looses atmosphere faster. And also make it with less of a magnetic field to protect the atmosphere from loss.
The highest known human town, La Rinconada in Peru, population 30,000, has an altitude of 5,100 meters or 16,728 feet.
This Wikipedia list: https://en.wikipedia.org/wiki/List_of_highest_towns_by_country1
Has nine countries with settlements over 4,000 meters (13,123.36 feet) elevation.
Thus there are tens or hundreds of thousands of humans today who have adapted to living above 4,000 meters permanently, although many people have difficulty with much lower altitudes. Thus if the pressure at sea level of your planet was the same as at 4,000 meters on Earth, native intelligent beings could easily have evolved to flourish at that pressure.
At 4,000 meters above sea level the air pressure is only 0.85 that at sea level.
The atmospheric pressure on the Martian surface averages 600 pascals (0.087 psi; 6.0 mbar), about 0.6% of Earth's mean sea level pressure of 101.3 kilopascals (14.69 psi; 1.013 bar). It ranges from a low of 30 pascals (0.0044 psi; 0.30 mbar) on Olympus Mons's peak to over 1,155 pascals (0.1675 psi; 11.55 mbar) in the depths of Hellas Planitia.
These pressures correspond roughly to pressures on Earth at altitudes of 30,125 meters or 98,350 feet to 57,150 meters or 187,500 feet - 30.125 to 57.18 kilometers or 18.626 to 35.511 miles.
Even if the air pressure inside your sinkholes is only the same as at 4 kilometers or even 5 kilometers on Earth, and even if the air pressure of Hellas Planita is defined as the Martian air pressure, that would still reduce the needed elevation difference by only 4 or five kilometers to 25 or 26 kilometers or 15.534 to 16.155 miles.
On Earth the difference between Mount Everest 8,848 meters, and the Challenger Deep, minus 11,034 meters, is 19.882 kilometers or 12.3541 miles.
Because the Earth is an oblate spheroid, the difference between the farthest point from the Earth's center, the peak of Mount Chimborazo, at 6,384.4 kilometers, and the closest point to the Earth's center, the bottom of the Litke Deep in the Arctic Ocean at 6,351.61 kilometers, is 32.79 kilometers or 20.1 miles. But the atmospheric density would tend to be equally oblate as the Earth's surface instead of spherical, i think.
Because rock on the sides of slopes a has a direction without presssure, it can collapse and slide down the slopes while rock inside a mountain has no place to go. Thus mountains and other elevation features tend to collapse and avoid reaching heights above the limits of their structural strengths.
Thus it seems very unlikely that a planet similar to Earth would have a much higher range of elevation, sufficient for the one lowest spot on the surface to have a breathable atmosphere and for the one highest spot on the surface to have an atmospheric pressure as low as the highest atmospheric pressure on Mars.
And of course it would be many times less likely for a planet to have the surface pressure of Mars, even the highest surface pressure of Mars, over the vast majority of its surface, and have several depressions deep enough to have breathable air at their bottoms. Stone just isn't strong enough for such depressions to avoid collapsing and filling in.