Sir,
I became unconscious, then I fell down. So can I say?
I fell down became unconscious. Likewise
I felt giddy, then I fell down.
I fell down felt giddy.
Thank you.
It's very important to communicate whether you became unconscious before you fell down (causing you to fall down), or whether you became unconscious after fell down (perhaps as a result of a head injury).
"I fell down and felt giddy" doesn't communicate either of these.
"I fell down because I was giddy" is better. But why not simply say what you mean: "I became unconscious, then I fell down" -- except that this must be reported evidence unless you say, "I became unconscious, and then I must have fallen down, because I woke up on the ground."
Note: feeling giddy is not becoming unconscious, though it can lead to it.