Motors increasing slowly in Attitude mode is normal when on the ground or with props off. When Attitude (bank angle) is a little wrong, it speeds up some motors to make the Attitude correct. Because it is not in the air, it does not move, so it tries harder and harder. If props are on and it is flying, then the first smallest motor change will cause the Attitude to change and it will not have to spin up faster and faster.
In Rate mode it does not care that it is not perfectly level so it does not spin the motors faster (a simplification of what actually happens).
I think maybe that you don't actually have a problem, but you are not used to controlling the quad throttle.
The first thing you must learn is how tiny tiny changes (like just 1mm) in the throttle is the difference between slowly (which is what you want) climbing and slowly descending. In order to keep it at waist high, you will always be adjusting the throttle up and down by the tiniest amount possible, just like when driving a car on a straight road, you use the smallest amount of steering to keep it centered in the lane.
As long as your quad is not super super powerful then reducing the throttle to say 1/4 should cause it to descend (with motors still turning); and as long as your quad is not weak, then increasing the throttle to 3/4 should make it climb. The key here is to find the place where it is not climbing or descending and from there, only use maybe plus or minus 2mm of throttle stick movement. If you use more, you over control. As @f5soh said, do this outside until you "get the feel of it" and can control altitude.