I transcribed these excerpts from sound samples found at Yahoo! Music.

Here's Gene Autry's song:

...to blossom in heaven above.
There are millions who miss you, friend Jimmie,
Miss that soft voice that made millions happy,
May God in his mercy grant your friends their one prayer,
That they'll meet...

And here's Bradley Kincaid's song:

...sorry that you had to go
But on records we'll hear you forever
You sing songs that we cherish so.

When you sing in that heavenly choir,
Won't you sing "You and My Old Guitar"
And your sweet song, "Blue Yodel..."

The tune of Kincaid's song is clearly "Red River Valley," but Autry's is a tune I don't recognize. Oddly, it doesn't seem to rhyme.