Los Angeles Airport

Have you ever transferred to a different flight at in Los Angeles International Airport?

I'll take a flight from Vancouver to L.A. via Alaska Airlines with two pieces of baggage. (When will you take the flight - on which day?)
It will be arrive at 8:55 AM, and I'll transfer to the next flight from L.A. to New York via (or "on") American Airlines at 11:00 AM.

Since the next flight is on a different airline than with the first one and it doesn't even include free baggage, I'll be worried if it takes a long time.
I will have to check if I have to go through immigration and take (or "recheck") my baggage at LAX or JFK airport in the airport by myself.

The travel agency, which made a reservation for those flights, said that I would not have a problem.
But if I feel embarrassed at the airport, I won't be able to speak English fluently, and it will result in a longer transit time.

The only thing what I want to do now is just to check with the a customer service each airline desk at the L.A. Airport.
I hope that I can transfer without any difficulties.

(Excellent work!  You wrote an interesting entry with very good grammar.)