I’ve had 2 lucid dreams so far and both of them ended with a false awakening, both of them very realistic. Is this something I should expect after most LDs? Should I start doing RCs right when I wake up in the morning?
YES! FA will happen extremley frequent if you try to ld a lot (at least for me, I’ve had at least 15 in the last 2 weeks) And they happend a lot after lucid dreams. So do those RC’s every single time you wake up, so you will also do them when you actually didn’t wake up 
YESS! As the other member above me said do them every time you wake up because they happen a lot more than you would think.
And I have had a FA which was actually a lot of fun but I can’t go into details for a few reasons 
Good Luck
I don’t usually get FAs, but when I do I usually get a long string of them. I remember one morning I “woke up” after my LD, went upstairs, ate, talked to my family, etc, then I woke up again. Like yours, it was extremely realistic. Anyways, I “woke up” and lived my morning over twenty times that morning, until finally when I did actually wake up, I just laid there for a while just waiting to “wake up” again, thinking it was another FA. Those were my first FA experiences that I can remember, and it was kinda freaky. But now FAs don’t bother me at all 
Someone in another thread (I forget who) made a really good suggestion about FA’s.
If you keep a dream journal near your bed, put a piece of paper near it with “Are you dreaming?” written on it in large letters. Always keep it there near your dream journal, and do a RC whenever you see it throughout the day to create that habit.
That way when you wake up and experience a FA, you’ll see the writing and automatically do a RC and discover you are, in fact, dreaming
Almost all my FA’s involve me waking up and writing in my dream journal, only to wake up for real later on and discover the page is empty 
That’s a fantastic idea 
Thanks for sharing Troober