I know this doesn't apply to today's flus or flu vaccines, but about 8 years ago I got the seasonal flu shot. A few days or a week later, I came down with the worst flu of my life... I could barely get out of bed for days. I have no idea if if I "caught the flu" from the flu shot or if the vaccine just didn't do the trick...
I am hardly an expert and you should by no means take this as a fact but it is my understanding that it takes a least a week for the flu vaccine to be effective. I also understand that there is a lag time between the time you "catch" the flu and become symptomatic so there's a good chance (if the above is true) that you already had the flu when you got the shot.
I mention this because the same thing happened to me many years ago so I dropped getting the shot. Then after having the flu for a couple of years in a row I started the shots again, getting them as early in the season as they were available, and haven't had the flu since.
I understand your decision and by no means am I trying to change your mind, I only mention this just in case you want to do more research because yeow, those couple of cases of flu I had were brutal and it took me weeks before I got back to whatever passes for normal for me.
The flu shot cannot give you the flu, on the other hand, it takes a few weeks to build up the protective antibodies.
It sounds like bad timing.