Dr. Demento is actually more of a "Master of Ceremonies" presiding over the presentation of the bizarre/absurd via his radio show than he is a performer, though "Shaving Cream" is an exception where he does actually perform (on Dementia Royale).
As for "Funny Farm", the actual title of the song is "They're Coming To Take Me Away, Ha-Haaa!" by the immortal Napoleon XIV.
If you're really interested in finding Dr. Demento stuff, I'd suggest you check out:
You'll probably find more Dr. Demento related stuff there than you could even imagine, though the bits that are for sale seem awfully pricey to me.
A discography of Dr. Demento can also be found at: