Two things.  If you are accellerating when braking then either you are rolling the throttle open while squeezing the brake or as you said your thumb is hitting it.........just make sure your throttle cable still has the screw in the keeper on the bottom side of the throttle housing where the cable enters the housing.  Your thumb could be hitting the cable and rolling it out of the hosing causing it to accelerate

It does sound like the lever is too large a pull.  What year bike do you have?  The only suggestion I would have is that you might be able to change the master cylinder/brake lever out to one off a newer GS500E/F.  The had adjustable brake levers if I remember correctly.  Suzuki used two bore diameters on the master cylinder from the mid 80's through the 90's.  One for all the twin brakes and a smaller one for the single brake set ups.  The GS500E/F use the smaller bore just like the savage/S40.  I am building a cafe bike using the savage engine and have a mix of GS500E brakes and currently a savage master but may switch back to the GS500E master but they are interchageable as far as cylinder size.  The key to the cylinder size is it affects the lever throw and force required for braking.

If you want help in checking this out let me know via email and I can do some cross checking of parts for you.

I haven't checked in a while - but most of the time on ebay were some replacement levers - they are wider and closer to the bar so you don't need to reach for them - put set on the better-half's '02

  Try using the middle finger on the front brake then you don't pull the gas at the same time,I use mostly the rear brake at slow speeds.

When I first started riding I had that problem. What I was doing was keeping my thumb wrapped around the throttle and reaching for the brake with my fingers.

The fix? Make sure you take your thumb off when you reach for the brake.

-D. Dwarf

Also try adjusting your wrist position when you grab the throttle.  You may be reaching the brake but the throttle isn't fully rolled off.  This can happen if you reposition your hand while you ride, so you change where your hand is in relation to the brake.

Before you get into something more complicated:
Before you roll the throttle (you can check even without the bike running) can you reach the brake OK?

If so, position your hand where you can reach the throttle, and practice going to 1/4 or 1/2 throttle (best when on) and reaching the brake again.

One other set up suggestion.  When you sit on the bike with your arms to the bars in your natural riding position, straighten your fingers straight out like you are about to squeeze the brake lever.  Your fingers and arm should make a straight line.  Then check that the brake lever just rests under your fingers at that same level.  If you have to reach down (arm and fingers are angled to each other) then the lever is too low and the whole master cylinder should be rotated up.  if the you have to bend your fingers back to get over the lever then it is too high and should be rotated down.  It is amazing how many bikes get quickly set up at the shop and never properly adjusted.

good luck.