Mid-Acts dispensationalist here, and yes, we do believe only Paul's letters are applicable to the church today. (So when I say "we", I mean mid-acts dispensationalists)
The main verse we use is 2 Timothy 2:15:
Study to shew thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word of truth.
Here we are told to "rightly divide" the word of truth. To put it more simply, all of the bible is for us, but not all of it is to us. In order to determine who the author is writing to, you have to look at who they say they are writing to.
If you were to just open your bible and look at a verse, say Genesis 6:14, you might read: "Make thee an ark of gopher wood; rooms shalt thou make in the ark, and shalt pitch it within and without with pitch."
So, are Christians supposed to go out and build arks? Of course not. But why? If all the bible is for us then why wouldn't we follow this verse? It is because this verse is not to us. Looking at the previous verse tells us, "And God said unto Noah". Likewise, we should use the same logic for every other verse/book.
I'm sure you've heard of the "interesting" laws in Leviticus, such as: you are not allowed to mix certain fabrics. Do Christians today need to follow this verse? No, because that verse was written to those under the law, specifically, those under the dispensation of the law.
Let's move farther into the book, there is a highly quoted verse in Jeremiah 29:11 "For I know the thoughts that I think toward you, saith the Lord, thoughts of peace, and not of evil, to give you an expected end." You've probably seen it on plaques and sold around town. Again, mid-acts dispensationalists believe this verse is not to us because we search for whom the writer is talking to. If we look at the first verse in this same chapter, we see who this verse is talking to "the residue of the elders which were carried away captives, and to the priests, and to the prophets, and to all the people whom Nebuchadnezzar had carried away captive from Jerusalem to Babylon". That description does not fit any of us today, therefore that verse is not for us today.
As you can see, any book in the "Old Testament" is either written to those under the law, or in the dispensation of the law (or prior when reading Genesis).
Another thing we notice is that there is God's chosen people, Israel, and the rest of the world, Gentiles. Looking at the first 4 books written in the "New Testament" (Matt-John) we see that these describe the teachings of Jesus. When Jesus was teaching, everyone he talked to was still "under the law". We see this when Jesus says, "If ye love me, keep my commandments," when responding to the question from the rich man of "what must I do to inherit eternal life". We are also given a clue to Jesus' audience when he says in Matthew 15:24 "But he answered and said, I am not sent but unto the lost sheep of the house of Israel." Jesus' message was to Israel. And his apostles' message was to Israel: "These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: But go rather to the lost sheep of the house of Israel" (Matthew 10:5-6). Why did He only want to deal with Israel? Because salvation was of the Jews:"Ye worship ye know not what: we know what we worship: for salvation is of the Jews." (John 4:22)
We, today, are not Jews. We are gentiles. So why would we take mail addressed to Jews and apply it to Gentiles?
When we get to Paul and his letters, we see he is writing to us, the Gentiles. Romans 11:13 "For I speak to you Gentiles, inasmuch as I am the apostle of the Gentiles, I magnify mine office". And Romans 15:15-16 "Nevertheless, brethren, I have written the more boldly unto you in some sort, as putting you in mind, because of the grace that is given to me of God, That I should be the minister of Jesus Christ to the Gentiles, ministering the gospel of God, that the offering up of the Gentiles might be acceptable, being sanctified by the Holy Ghost."
Paul clearly tells us who he is writing to.
Looking at Hebrews - Jude, those books were written to "the little flock" or "12 tribes" who are believers during the 70th week of Daniel (See James 1:1) based on the context clues/verses within the books. What I mean is that there are verses that "contradict" what Paul says. But we believe the bible does not contradict, so those books must be talking to different people during a different dispensation. These books talk about "enduring to the end" and "obeying the law" and "if you sin then God is not with you". These are all signs of the New Testament, as described by Jesus and the prophets from the Old Testament. A time when God will write the law on their hearts.
Revelation describes future events, and isn't really doctrine. So, it's hard to say it "applies" to anyone specifically. It's just telling future events, not ways to be saved that should apply to your daily life.
References to show that this isn't only me: