I would check out “The Good News about Sex and Marraige” and “Introduction to Theology of the Body” both by Christopher West. They should help in understanding why the Church teaches as it does.
As a recent converts my wife and I are having a good deal of difficulty in this area. For me, one of the things that drew me home was the focus on intent. To me this is very Biblical and is found throughout the Church’s teachings and those of the ECF… except on contraception.
When Jesus said if a man lust after a women in his heart then he has already committed adultry, He is talking about intent. If your intent is to space children, which is ok, then why not use contraception? Yes God provided a women natural cycle to help us to that but the intent is the same, regardless of the method. To me if you are going to truly be open to life, then you don’t need to practive NFP because it’s the same “intent” as using contraception.
So I’ve said it…

I have a lot of other stuff I won’t voice but I will say that God has put it on my heart to better understand why the Church teaches as it does so I have West’s books as well as the new translation of JPII’s sermons… and we are praying about it as well. I accept the teachings of the Church, it’s just difficult to change 38 years of understanding. With God’s help though and His grace I’m sure we will make it
Now back to the OP, I wouldn’t be concerned about your wife being 38. We too tried (probably due to years of oral contraceptives - I was referring to condoms above, we would never consider oral again due to the aborficient properties of many of them) for many years. Initially seeking medical help but getting pregnant when God helped us to be ready for children. We now have 3 boys (5, 3, and 1). I’m 38 and my wife is 36 but are open to more when the time is right should God choose to bless us with one (maybe a girl this time).
I know many people into their 40s having kids with no problems so don’t let that worry you. Rather focus on the great gift you have been given (I’m sure you are both still in awe

) and work on understanding why our faith still teaches what the other Christian churches have abandoned, that contraception is morally wrong.
Remember that ALL Christian’s were taught that it was wrong up until the 1930s.
I spelled out our stuggle above so that you would now you weren’t alone. A lot of people here have been taught this or understood it all their life… we haven’t so it’s difficult. Yet we have been so blessed, as you guys have too, we want to do what He desires. Study and pray and I’m sure He will open your eyes, I have faith that He is going to enlighten us as well.
Joe