Drinking and eating on the bike

Before reaching for a bottle, get your speed up a bit,  pedal smoothly and move one hand (usually your non-dominant hand) to the bar tops near the stem and keep a light grip there. Now reach for the bottle with your dominant hand, while either coasting or still pedaling smoothly. With your hand near the stem, it takes more hand force to cause the front end to wobble. You'll be more stable. Same goes for reaching into a jersey pocket -- move that non-dominant hand to the bar top.

Yes.  Easier to do when going a bit faster or even easier if you are coasting safely downhill.  You don't want to reach for anything if you ever need to touch your brakes though...so stress on "safely downhill."  The faster you go, the more stable you are.  The hardest thing to do is reach for bottles while climbing uphill and you need to hold your line.  Usually not a problem on training rides or triathlon races but harder to do when climbing a hill in a big pack.

You are most likely drinking too much prior to getting on the bike. That's what often leads to my wobbling. 

I absolutely think this is something you can learn with time.  Also, poor core strength really makes you push the handlebars too hard and when you remove one hand.  Try to hold yourself up with your core and not your hands.  I see myself do this in the early season a lot.

That said, you have a big race in 6 weeks.   I personally prefer to wear a hydration backpack (Camelbak, whatever) for long rides.  I did this in my HIM race.  It's not very "cool" but I was able to carry my entire fluid & calorie requirement without stopping and could drink along the way.  I felt totally in control.

Just a thought...