First, it IS possible you got it from the birds - sure, it
can happen.
However, it is possible you got it from lunchmeat or other processed type meat you got from the grocery. Even ice cream or other dairy products can be the source. Look up sources of Salmonella and there is a whole list of common sources of the infection.
Glad you actually have a diagnosis, so you do, in fact, know you were ill from Salmonella and your nor doctor are speculating.
There have been Salmonella outbreaks linked to backyard birds...which were linked back to a hatchery. For example (not the only one), Mt Healthy Hatchery in Mt Healthy, OH (near Cincinnati) has been linked to 2 outbreaks in different years according to CDC info (pretty sure this is on the CDC website...maybe it is another website as I read this information a year or so ago). However, all birds do not have Salmonella, and all hatcheries do not have a problem.
You do not have location listed on your profile. It can be a general location or a state - but it can help with responses. Each state has a place to send a dead bird for necropsy. The Emergecies forum has helpful links, but does not include a state listing of labs for necropsy. I think
@Kiki might have been the one to create a list of state labs that can receive and evaluate a dead bird, but I'm not sure. Of course, each state is different in cost, with some being a reasonable cost, others are more costly - but likely cheaper than culling your whole flock needlessly.
Two links for summary info on Salmonella:
CDC info on sources of infection:
https://www.cdc.gov/foodsafety/communication/salmonella-food.html
Poultry DVM website with chicken specific info
http://www.poultrydvm.com/condition/salmonellosis
Good Luck.