Latter Day Saint doctrine is clear that eternal marriage exists in the Celestial Kingdom. Is there doctrine stating it won’t exist in the other two kingdoms?
Yes. It is found in the same revelation as that on eternal marriage, though it may require some deduction to understand. First, we should note the teaching in D&C 131:1-4 that in order to attain the highest degree of celestial glory, one must enter into the new and everlasting covenant of marriage. If one doesn’t, and yet is otherwise worthy of celestial glory, one will not attain its highest degree.
Next, we need to look at a few verse in D&C 132. First, part of verse 7, and verses 8-10:
7 … All covenants, contracts, bonds, obligations, oaths, vows, performances, connections, associations, or expectations, that are not made and entered into and sealed by the Holy Spirit of promise, of him who is anointed, both as well for time and for all eternity, … are of no efficacy, virtue, or force in and after the resurrection from the dead; for all contracts that are not made unto this end have an end when men are dead.
8 Behold, mine house is a house of order, saith the Lord God, and not a house of confusion.
9 Will I accept of an offering, saith the Lord, that is not made in my name?
10 Or will I receive at your hands that which I have not appointed?
This is saying that any marriage entered into other than those sealed by the Holy Spirit of Promise and performed by one who has been anointed to this authority ends at death. The Lord will not accept any agreement which he has not appointed. Verses 13 and 14 emphasize that whatever is not ordained of the Lord will be destroyed. Then, in verse 15-17, the Lord explains this explicitly in relation to marriage:
15 Therefore, if a man marry him a wife in the world, and he marry her not by me nor by my word, and he covenant with her so long as he is in the world and she with him, their covenant and marriage are not of force when they are dead, and when they are out of the world; therefore, they are not bound by any law when they are out of the world.
16 Therefore, when they are out of the world they neither marry nor are given in marriage; but are appointed angels in heaven, which angels are ministering servants, to minister for those who are worthy of a far more, and an exceeding, and an eternal weight of glory.
17 For these angels did not abide my law; therefore, they cannot be enlarged, but remain separately and singly, without exaltation, in their saved condition, to all eternity; and from henceforth are not gods, but are angels of God forever and ever.
In verse 18, the Lord says that even if they make their covenant for time and eternity, if it is not authorized by him or sealed by the Holy Spirit of promise, it is no longer valid once they are dead.
The Lord has not established any form of marriage other than the new and everlasting covenant, which is a celestial law and if sealed by the Holy Spirit of promise, allows those so sealed to enter into the highest glory of the celestial kingdom. Therefore, no one will be married in any other kingdom or degree of glory. Verse 17, in stating that those who did not abide this law will remain separate and single forever and ever, leaves no reason to suspect or hope that this may change.
I hope this answers your question, Shanna.