Some wobbling is normal for a new rider. If you are going in for a fit, have them check to make sure your stem is not too short- this can cause excess wobble in your steering.
You will also likely find that at first your body might seem too stretched out and far fromthe handlebars....but with riding time your core muscles will strengthen and the reach will seem less extreme and more comfortable. Your stronger core muscles will also help keep you from feeling like you are falling forward onto your hands, because you will begin to distribute your weight and balance more evenly between seat, hands, and feet.

You want your "cockpit" room to be neither too long or too short.
Sometimes new riders feel the need to shorten the reach a lot by doing things like putting shorter stems, pushing saddle all the way forward, tilting handlebars up, raising the handlebars...etc...and then months later with stronger muscles they feel cramped and need to lengthen it again to a more "normal" fit and reach. Keep in mind that our bodies change and adapt with time in addition to the changes we make to our bikes.