Potty training occurs in 3 stages:
1) peeing in toilet and mastery of it.
2) pooping in toilet and having mastery/ability for it
3) night-time dryness. This can take even up until 7 years old. This is normal. This is per our Pediatrician. Many Kindergarten kids as well, still wear night time diapers.
your son is has peeing learned. He still is going through pooping learning. This is normal. He is young.
He will NOT have night-time dryness occur at the same time, just because he is peeing in the day-time, in the toilet.
"Diapers", the night time kind, does hold more. Its okay. He has not attained this phase yet. It is a biological attainment.
My daughter, was wearing night time diapers, not pull ups, until she was 5 years old. Normal.
For leaking accidents, use a water-proof bed pad directly under him. To make clean up easier for you. You can get it at Amazon. They have all kinds. This is where I got mine. I have about 4, which I use for my kids, and just switch them out and wash them as needed. No biggie.
Keep in mind, that "night-time" accidents will still occur, even if they are night-time dry and have the ability. My daughter, is 7 now. And she once in awhile still has accidents every blue moon. She even dreamed once that she was on the toilet, until she felt herself wet. It was funny and we just laughed about it. It wasn't her fault. She's just a kid. And she sometimes sleeps so deeply, she does not wake to go pee.
Don't put onto him, that you are disappointed in him. He is normal. He is STILL learning about toileting. He is still young.
Many kids, when still getting used to potty training, will poop in a diaper only. This is the process of it.... then one day he will be okay about going in a toilet. Its okay. Normal. MANY kids do this. It is a process.
All kids, have their own pacing about it. My son is 3, and still wears night time diapers too. I wake and change him 1 time a night... so his diaper does not get like a swimming pool. Its okay.
And, you also have a 4 month old baby. HE has a new baby around too, to get used to. SO MUCH on his plate, to adapt to. Let him. At his pace. When a child is "stressed" they can often even regress. Its normal. Don't pressure him or think about the cost of pull-ups for now. A child, can't fathom that.
Your son is still learning. He is daytime pee trained now. He still has to learn the pooping and much later, night-time "dryness" will occur. They say that boys are sometimes later in attainment of toileting.
All the best,
Susan