The verb fawn is not complimentary. Dictionaries define it as making a servile display of flattery or affection.
The Dublin singer-songwriter [James Vincent McMorrow] had critics fawning over him when he released his debut album in 2011.
Daily Mirror, 3 Jan. 2014.
It didn’t start out like that. A thousand years ago it was applied only to dogs, who showed their delight by whining or wagging their tails. The word is Old English, from fægen, to rejoice or be glad. It was a special case of fain, to be glad or pleased, which went out of use in the sixteenth century, leaving the adjective, which itself is now obsolete. My earliest memory of adjective fain is from the ballad of Lord Randal: “For I am weary with hunting and fain would lie down”, meaning that he would be very pleased to be able to rest.
Fawn stayed in the active language, though the idea of a fawning dog was long ago applied with greater force of insult to a human who acted like one.
Spoken like a true dog. A fawning, slavishly affectionate, drool-dripping dog who’ll cut off his left ear in return for a pat on the head.
Kingdoms of Light, by Alan Dean Foster, 2001.
Incidentally, fawn for a young animal, particularly a young fallow deer, derives ultimately from Latin foetus, offspring. The colour comes from that of the animal’s coat.