There can be very simple answers to your questions. But the more experienced folks here seem to jump to conclusions and make it complicated. Maybe because they enjoy the process and it's embellishments, rather than seeing it as a means to an end that should be pragmatic and minimized.
Says the man who gives the longest most complicated post in the thread.
Please excuse daft, his comments sometimes are a little daft and/or inflammitory.
Of course we enjoy the process. Why else would we be doing it? If we didn't, we'd go to the store and buy soda or beer, because that's easier and faster.
I didn't see anyone recommend in this thread to jump in whole hog and buy a kegging setup or even waste money on a silly little sodastream. Roadie simply offered a straightforward explanation of what force carbonation is and how it's typically accomplished in homebrewing.
Also, I simply offered that maybe a yeast nutrient might help. I never said the yeast type was absolutely the problem. I see Nogud247's suggestion to try a champagne yeast as a simple solution since champagne yeast is so inexpensive and I have definitely seen a difference in yeast strains, even in a short soda fermentation. If you don't believe me, try fermenting straight up grape juice with Red Star Pasteur Champagne versus Red Star Premier Cuvee, then tell me there's no difference. And yes, to a lot of people, a "yeasty" flavor is similar to a "beer" flavor, so that's how I interpreted the question.
SodaCoastie, I apologize if I misinterpreted your description of what you thought was off. But to answer your question, a brewers yeast probably won’t help. I think if the yeast flavor is off putting, then it’s probably not your preferred method of carbonating. You could try making a syrup to add to carbonated water. Use your same recipe, but heat the sugar in a little bit of water to get it to fully dissolve. Put that in the bottle first, then carefully fill with cold seltzer or club soda. Pour it down the side as you fill so it's a more gentle process. You'll keep more of the carbonation that way. The key is to keep it from mixing or sloshing much until you've sealed the bottle back up. Once you've sealed, shake until dissolved and then let it sit and equilibrate for a few minutes before you open it back up again. If that get's you closer to what you expected, then you know your recipe is on the right track. I would agree with daft that you may want to add some tartness, but I disagree that you'll get it from a slug of yeast. Try a touch of lemon juice to see if that helps. It does seem to me that maybe your vanilla extract is too much, but I don't think the extract itself is the issue. I think you'd be ok with half that amount. Cream soda is typically flavored with vanilla extract and you're on the right track using pure vanilla instead of any imitation stuff. For a more complex flavor, you might try light brown sugar instead of white. Good luck to you. I hope it turns out great!