Stabilisers are usually used when turning between centres not for bowls.
If the problem occurs on every job you try, you may have bearing failure and will need to investigate further.
More likely is that the blanks and WIP aren't balanced and are generating the vibration. Making sure the lathe is tightly bolted down onto a really solid base is a vitally important first step.
Once you've rounded off the initial blank things should calm down a lot, but it may take a while with careful light cuts to round things off properly.
Ensure that the work piece is really secure on the chuck (or other mounting). You can smooth things out carefully then if the mounting isn't secure one heavy pass or maybe a small dig can knock everything out of balance again.
It's also worth remembering that some pieces, natural edged work or with burrs in, may never be completely in balance anyway. So you may have to live with some vibration even if the lathe set up is perfect.
The Avon is a pretty good lathe (I've got one myself and have spent the morning fighting an out of balance blank) but make sure everything is tightened up very well when working with out of balance work, vibration can make taking accurate cuts to round things off more difficult. Keep rests as close to the work as possible to reduce any overhangs that allow any vibration to be amplified.