After reaching out to Pressman Toy regarding the ruling I received the following:
In the Oregon Trail narrative, players throw supplies overboard to help cross the river. But, even though the wagon may become empty and technically it would be easier to float, they are stuck. Players must continue to follow the instructions until someone finally does roll ...
In the SAMPLE PLAY section it says:
Bob says, "I have a Clothes Card I can spare for you."
This hints that you can share information about what cards you have before you play them. Frankly I don't see how it is winnable without some supply card teamwork.
A Board Game Geek user sent in some rules questions to Pressman Toys and received responses from the company president and game designer. They were kind enough to post them on Board Game Geek, and this question was one of them:
1) River card: “Roll an Even number to ford the river. Roll a 1 and die by drowning.” What happens then when the person rolls a 1 ...
Your oxen count is not based on the number of supply cards you have - it is as you mentioned that you just have an automatic 2 oxen. When calamity cards affect your oxen they will just affect/kill all of them, and a single oxen supply card can save you from being stranded without new ones.