"The Enchanted Tsarévich" is an ATU 425C: Beauty and the Beast tale from Russia. This tale also describes the beast as a snake, but a winged one, perhaps more of a dragon image although snake is used in the translation by Leonard Magnus from Alexander Afanasyev. It is included in Beauty and the Beast Tales From Around the World.
Here's the opening of the tale:
So he ran up and broke it [a flower] off, and as soon as he had done it, in that very instant a boisterous wind arose and thunder thundered, and a fearful monster stood in front of him, a formless, winged snake with three heads. “How dared you play the master in my garden!” cried the snake to the merchant. “Why have you broken off a blossom?”
The merchant was frightened, fell on his knees and besought pardon.
“Very well,” said the snake, “I will forgive you, but on condition that whoever meets you first, when you reach home, you must give me for all eternity; and, if you deceive me, do not forget, nobody can ever hide himself from me. I shall find you wherever you are.”
This version of the tale does not vary too much from a standard Beauty and the Beast, except in the sleeping arrangements. There is also no nightly marriage proposal:
Darkness now came on, and the merchant’s daughter went into the bedroom, wishing to lie down and sleep. Then a boisterous wind rustled round and the three- headed snake appeared in front of her.
“Hail, fair maiden! Put my bed outside this door!”
So the fair maiden put the bed outside the door and herself lay on the bedstead.
She awoke in the morning, and again in the entire house there was not a single soul to be seen. And it all went well with her. Whatever she wished for appeared on the spot.
In the evening the snake flew to her and ordered, “Now, fair maiden, put my bed next to your bedstead.”
She then laid it next to her bedstead, and the night went by, and the maiden awoke, and again there was never a soul in the palace.
And for the third time the snake came in the evening and said, “Now, fair maiden, I am going to lie with you in the bedstead.”
The merchant’s daughter was fearfully afraid of lying on a single bed with such a formless monster. But she could not help herself, so she strengthened her heart and lay down with him.
In the morning the serpent said to her, “If you are now weary, fair maiden, go to your father and your sisters. Spend a day with them, and in the evening come back to me. But see to it that you are not late. If you are one single minute late I shall die of grief.”
I would love to see some illustrators' images of this "formless, winged snake with three heads," wouldn't you? There are perhaps some out there, but I didn't find any with a cursory search.