Hello and welcome to Flyertalk!
I am a volunteer moderator for this board. We are all travel fans, but (with only a few exceptions) do not represent airlines in a professional capacity. There is almost no chance the airlines will see your post on Flyertalk. You need to contact the airlines to get the answers to these questions. Most airlines use something called Timatic to determine whether passengers are eligible to fly to specific countries with or without certain visas. When you check in at the airport, the employees will reference Timatic or a similar software to determine whether you're allowed on the plane. It is possible to check this information in advance on your own, your best bet is to contact the airlines to confirm.
Also -- and I suspect you've already realized this -- this is why cheap/low-cost travel agents may actually be more expensive in the long run, since there is a chance BudgetAir sold you non-refundable tickets that cannot be used by someone with your specific passport. I hope that's not the case.
chgoeditor
co-mod, Info Desk