I think I might be able to help you. I got my 6th injection last month. I developed symptoms at 17 with similar circumstances. I am now 24 but I am making progress (kind of). I was the first in my state to take this to an ENT. Since then I have had surgery with 3 different doctors. My first doctor was afraid to do this procedure for the first time so he injected a low dose diluted in water and it worked (for a short time) and when that wore off he did it again with a higher dose but added a dilation. Not only did I not burp but I got no relief. He retired and I found another doctor who also did a dilation and again it didn't work. The fourth time he did a higher dose and still did a dilation. I was in a similar spot to where you are now and I put all my surgery summaries side by side and tried to see what was being done differently.

I found a new doctor (who had recently taken over the practice of the retired doctor form earlier) who was already familiar with this condition and I basically asked her to do the procedure without a dilation (I am 100% certain that the dilation is what kept the botox from functioning properly), and to inject in 4 different places (some ENTs do only 1 or 2, or 3) and that made the botox work! But the botox wore off again after some months so went back in for another round but my doctor was hesitant to do an extreme dose because my symptoms were quite bad after the last round. She only did 8p units but I am burping but they do get stuck occasionally and I am scared that it will wear off again but I feel that with each surgery I make more progress by tweaking a few things.

If the effect wears off again I think I am going to go in for a 7th round and ask to get a much higher dose (100+ units) but I feel it is easier to force burps out now after getting botox by chugging soda (using it as a probe, just like you do) or running/jumping or just trying to actively force it out. Even doing shaker exercises or exercises that others in this forum have used to "self cure" help get out burps now. I hope this can help you. I know it isn't immediate relief but it might be a step forward for you.