Hi K.,
I just asked the PED about my daughters thumb sucking because she has developed sores on her thumbs too. My daughter will suck either thumb and sometimes tries to get both in her little mouth.
The doctor said we can't put medicine on her thumbs for the sores because she will put it in her mouth. The sores have gone down, but we still have the issue of thumb sucking. He said to just remind her.
So, depending on the day and how tired she is, I have talks with her about sucking her thumb. I remind her repeatedly to take her hands out of her mouth. Sometimes she will just take it out and sometimes she will grunt at me and turn her head. My daughter is 18 months, but I am sure your 1 year old will understand after gently removing her thumb a few times.
C.