If one were to pick and choose what portions of the Bible to believe or not believe, everyone would end up choosing different parts and in no time at all, if everyone 'put their heads together', they'd throw the Bible in the trash. Perhaps the pick and choose 'theology' is due to ones' upbringing, teaching, and even the church and preachers included. Then one has to discern whether the preacher/Sunday school teacher is rightly dividing Gods' Word or are they misinterpreting something. One can only discern the difference through knowing scripture. And one thing I've learned through the years reading and being taught Gods' Word by very Bible-knowledgeable preachers and teachers at church, the more I know of His Word, the easier it is to 'connect the dots' of what is being taught to the rest of His Word.
I hesitate in bringing up the term: dispensation, as I know there's more than a few anti-dispensationalists on this forum. There are dispensationalists that will argue how many dispensations are in the Bible. It's easy to think of dispensations like raising a child. As a toddler, we know they are learning much and we're certainly not teaching them calculus before they can even count to 10. As teens, we reinforce what they've been taught in their early years, and add to that how to drive a car, the basics of dating, and in school, they get some higher level math, science, etc taught to them. I'd call it a dispensation of cash to put them through college. In short, as parents, we both teach and treat our children differently as they grow up and eventually go out on their own...although we're still concerned about their welfare and may help them if needed.
Knowing what dispensation God is using dealing with individuals in the Bible is important, particularly the Old Testament. Dispensations are serial, where one ends, the next one begins. But there is sometimes a brief overlap between them when necessary. Adam and Eve were in one dispensation (innocence). There were several intervening dispensations until the dispensation of law through Moses. Once Paul started his apostleship to the Gentiles beginning with his 3 years in Arabia being taught by Jesus, the dispensation of grace officially began. Prior to Jesus and Paul in Arabia, the age of grace (aka dispensation) was completely hidden from everyone. Even from Jesus' disciples. He even told them He was going to Jerusalem to fulfill all prophecy about His first coming and they understood none of it. Gods' going to the Gentiles through Paul was kept hidden as well.
Luke 18:31-34 (KJV)
31 Then he took unto him the twelve, and said unto them, Behold, we go up to Jerusalem, and all things that are written by the prophets concerning the Son of man shall be accomplished.
32 For he shall be delivered unto the Gentiles, and shall be mocked, and spitefully entreated, and spitted on:
33 And they shall scourge him, and put him to death: and the third day he shall rise again.
34 And they understood none of these things: and this saying was hid from them, neither knew they the things which were spoken.
The dispensation of grace actually overlapped the dispensation of law. Paul started his ministry to the Gentiles roughly 40 AD, but the temple continued (dispensation of law) continued until its destruction by the Romans in 70 AD. At that point, praying at the temple, the temple priests, animal sacrifices, etc ended...ie, the dispensation of law ended. Until Paul came along, salvation by grace through faith alone was completely unknown. Were the Jews that believed that Jesus was the promised Messiah saved because of grace? Nope. During Jesus' ministry, Jews only needed to believe that He WAS the promised Messiah. Period. Just like Peter told Jesus -
Matthew 16:16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God. (KJV)
Nothing about Jesus death for our sins, burial and resurrection. Just faith that He was the Messiah.
Only Paul taught salvation through grace by faith alone. Of course, our fully believing that Jesus IS the Son of God that paid for all our sins at the cross is first and foremost. His death was the perfect sacrifice for sins, exactly as animal sacrifices for sins were made under the law.
Ephesians 2:7-10 (KJV)
7 That in the ages to come he might shew the exceeding riches of his grace in his kindness toward us through Christ Jesus.
8 For by grace are ye saved through faith; and that not of yourselves: it is the gift of God:
9 Not of works, lest any man should boast.
10 For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them.
Paul all about grace. Not law. And wrote of our not being under the law, but of grace.
Romans 6:14-18 (KJV)
14 For sin shall not have dominion over you: for ye are not under the law, but under grace.
15 What then? shall we sin, because we are not under the law, but under grace? God forbid.
16 Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness?
17 But God be thanked, that ye were the servants of sin, but ye have obeyed from the heart that form of doctrine which was delivered you.
18 Being then made free from sin, ye became the servants of righteousness.
Just to clarify the dispensation grace, it was kept hidden (eg, a mystery) until revealed to Paul -
Romans 16:25 Now to him that is of power to stablish you according to my gospel, and the preaching of Jesus Christ, according to the revelation of the mystery, which was kept secret since the world began, (KJV)
Galatians 1:11-12 (KJV)
11 But I certify you, brethren, that the gospel which was preached of me is not after man.
12 For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.
It is Paul alone that was taught and teaches the dispensation of grace -
Ephesians 3:1 (KJV)
1 For this cause I Paul, the prisoner of Jesus Christ for you Gentiles,
2 If ye have heard of the dispensation of the grace of God which is given me to you-ward:
3 How that by revelation he made known unto me the mystery; (as I wrote afore in few words,
4 Whereby, when ye read, ye may understand my knowledge in the mystery of Christ)
5 Which in other ages was not made known unto the sons of men, as it is now revealed unto his holy apostles and prophets by the Spirit;
Colossians 1:24-27 (KJV)
24 Who now rejoice in my sufferings for you, and fill up that which is behind of the afflictions of Christ in my flesh for his body's sake, which is the church:
25 Whereof I am made a minister, according to the dispensation of God which is given to me for you, to fulfil the word of God;
26 Even the mystery which hath been hid from ages and from generations, but now is made manifest to his saints:
27 To whom God would make known what is the riches of the glory of this mystery among the Gentiles; which is Christ in you, the hope of glory:
Hopefully, all the preceding helps clarify Paul's writings. He makes it clear that being under the law (Exodus-John) is NOT the same as being under grace (Romans-Philemon).