1) Feed on demand, 24/7, if you are not already.
2) A baby this age typically gets "tired" after about 2-3 hours of "awake" time. So gauge his naps that way. And keep consistent about his nap times/routines. EVERYDAY have the same routine. This helps a baby a great deal.
3) he sounds either "over-tired", or did not get a "full" nap... and was woken before he could get back to sleep on his own and continue napping. (babies often stir or "seem" awake... but actually are not completely done with their nap. Try not picking him up right away, unless he is crying.... and see if he will go back to sleep).
4) rubbing his eyes, and moving his head back and forth... tells me that he is STILL tired and not done napping... this is a thing that MANY babies do... when they are tired. They do "rhythmic" movements to get back to sleep.
5) If he is gassy... use infant gas drops. That is what we used with our babies. It helps a great deal.
6) "filling up" a baby prior to bed, does not always "make" a baby sleep all night.
7) he is 4 months old.... at 3 months old, this is/was a growth-spurt time... and when they need increased intake to keep up with their growing body and rapid development.
8) sleep phases does occur... and so, it often just affect their sleep. Its normal.
9) Make sure you do have adequate milk supply.... or he will not be getting enough either. Also make sure he is latching on properly... or he will not be getting enough from nursing either.
10) some babies, will only nap well, in their crib/at home. Not on the road in a stroller. BOTH my kids were like that.
So I was always home at their nap times.
11) a baby this age, will not have a regular sleep pattern yet. It takes time, he is still so young.
12) Per my Pediatrician, for the 1st year of life, a baby needs to be fed on-demand 24/7, as this is their PRIMARY source of nutrition. NOT solids, and not other liquids.
13) "Sleeping through the night", for a baby, means sleeping at least 5-6 hours straight. A baby this age, will typically not be sleeping all night. They wake because they are hungry. This is normal. My kids woke like that too.
A great book is "What To Expect The First Year", which you can find at any bookstore.
Just some ideas,
All the best,