Mike, my perception of your problems is that the biggest problem you are having has more to deal with hard dry wood you are turning with variation in grain coupled with maybe holding/pushing too much on the bevel because of the bouncing you get when twice turning hard dry wood. It's tough to tell without watching you turn, but that is a common problem that I have until I force myself to relax my grip and take very small cuts until I have a smooth surface to guide the bevel.
Of course, check the lathe itself when you get the piece off it, but the fact that the lathe is new leads me more to the wood and perhaps tool presentation.
Mark made an important point about smoothing out the tool rest, little dings can cause lots of problems. I usually use a file to smooth out dings, but coarse sandpaper works as well. It's a matter of how deep are the dings.
As to getting rid of the center mass, I took a couple pictures of how I do that.
Position the tool rest so you are cutting just above center and point the bevel towards you at about 9:00. You are going to be cutting with the bottom edge of the tip and the bottom wing. Start at the center and work down towards the bottom while you are pulling the gouge towards you. Start with small bites (1/8") and take larger bites if the wood allows you to. I'm assuming of course you are using a gouge that you sharpen with wings.
My apologies if I'm telling you something you already know or have tried, just trying to cover all the bases. Wish I lived closer, a lot of times having a second pair of eyes can figure out a problem much quicker.