I remember there was a lengthy discussion about this on Introspective some years ago. As it happens, there are several songs called 'Love is strange', so which one Neil is referring to (if any in particular) is difficult to say.
A quick search on Google reveals songs by Mickey&Sylvia, Buddy Holly, Wings, Peaches&Herb, Kenny Rogers and Everything but the girl (though the latter two seem to written after 'I want to wake up'.
It was a cough that carried him off. It was a coffin they carried him off in.
EVERLY BROTHERS - LOVE IS STRANGE
(E. Smith; M. Baker)
Warner Bros WB 5649
Love, love is strange
Lots of people take it for pain
Once you've got it, you never wanna quit
After you've had it, you're havin' awful fits
Love is strange, love is strange
Hey, Don?
What, Phil?
How would you call your baby home?
Well, if I needed her real bad, I guess I would call her like this
Baby, oh sweet baby
My sweet baby, please come home
Yeah, that ought to bring her home, Don!
People don't understand
They think love is money in the hand
Your sweet lovin' is better than a kiss
When you left me, sweet kisses I miss
Love is strange, love is strange