If you're using Bill McCready's "Proper Method" and still weaving there are three things that come immediately to mind. No; make that four:
1. Make sure you're in a fairly low gear that's easy to pedal (e.g., 42 x 23 or 53/4 x 27)
2. Once you've got:
(a) your stoker mounted and clipped-in, and 3. At least for now, do not immediately attempt to clip in your down foot as you would on your single bike... let the tandem gain some momentum before you break the cadence and shift your attention to clip in.
(b) you've got yourself poised to launch with your preferred foot sitting on your crank in the forward position,
(c) you'll want to coordinate your launch by calling out, "Ready... on three: one, two, three" (or words to that affect) at which point,
(d) your stoker should begin applying moderate power to her pedals as you
(e) apply power to your pedal and while pushing off with your down leg and mounting your saddle.
4. Recognize that, like all things, your confidence and stability during starts and stops will improve with practice and familiarity.
The key is looking at the entire start and stop movements as a continuous flow of motion where both you and your stoker work to minimize any excess body movements that could unsettle the tandem's steering.
For example, if your stoker is nervous and trying to look around you, then she's unintentionally steering the tandem from the backseat which requires counter-steering inputs from you and in combination you end up with a wobbly path. Again, to minimize this during starts and stops you'll both need to become more relaxed with your weight centered on the tandem during your launches and landings and most of this should come to you with practice.
The teams that always struggle simply never master the finesse needed to have that smooth and relaxed start... or, don't pay attention to things like gear selection and technique.
Finally, and don't take this the wrong way, starting off in-phase would probably have been a bit easier at least for the first few rides, if only because it puts you and your stoker in-sync with regard to the natural side-to-side bike sway that comes during pedalling. Out-of-Phase simply adds a little complexity to the learning process. Once you have the starts and stops mastered -- at least in terms of timing and stability -- you cold then shift over to OOP, at which point you would only be sorting out the differences associated with being on different power strokes instead of dealing with everything else needed to master your start/stop techniques.