“I am planning to ask UC Davis about all this”
It is a good idea to ask UC Davis about this. However, I would expect that getting significant outside scholarships at this point is unlikely (as @thumper1 has also suggested), and I can’t imagine either UC Davis or the US government visa folks being unhappy about your applying for outside scholarships.
As I understand it (it has been decades since I applied for a student visa) you needed to show that you could afford to attend UC Davis to get the visa. However, “you can afford it” and “it is easy to pay for and seems cheap to you” are definitely not the same thing.
By the way, I am pretty sure that when you come to the US you need to bring your I-20 in addition to bringing your passport with the visa in it. I have heard of one case of a student who thought that the stamp in the passport was enough and left the I-20 at home. While this eventually got straightened out, it did cause some trouble.