I have the same problem with my shoes. The difference is I don't love mine and need to replace them so I didn't really care how I approached this problem. Be warned, you may not want to do this.

I too couldn't remove the faulty velcro so I took some velcro from my sewing kit, cut short of the full length (need room to move the strap around the buckle) then stapled the new piece onto the old non-tacky one. Now, the faulty strap was the one on top, the one that gets pressed onto the shoe. I wouldn't do this to the bottom, the staples might rub against your foot.

The new piece had adhesive so that helped make a closer contact between the two. I stapled from the inside of the strap because the staples on top of the strap really showed. Staple from the inside and all you can see are the little bent pieces on the outside. (My shoes are black and grey.)

I also used pliers to compress the folded/bent side. I used a regular desk stapler and it didn't have the best pressure for such thick material, but it worked once I used the pliers.

Its a very white-tape-to-hold-your-glasses-together type of solution but it was all I had at the time and it ended up working for a long time.

Oh, just saw Blueberry's solution. That sounds better for the shoes you love.