Mark, biennialism in apples is not so much a genetic trait (although some varieties seem prone to it, it's true) as a cultural one. Your new branches from a favored tree should be well-behaved if you manage them well.

First off, thin your tree ruthlessly. On my Liberty I have to thin five or six apples from each cluster to get down to one apple, and then I often remove half of the remaining apples. Second, thin early- right after fruit set. I think some people even remove some of the blossoms before fruit set.

I hope that helps.

Good luck,

Mark

Grafting a biennial bearer onto an annual bearer won't change bearing habit. You'll need to do as Mark suggests in order to get apples every year. Or at least to improve chances of annual bearing.