J.V.
Now is when you have to teach him how to put himself to sleep, and help the naps to consolidate. One trick that might work: 5 minutes before he usually wakes from a nap, go in, make a little noise, enough to get him to roll over, not wake up. It should reset his sleep cycle.
Also, make sure he is ASLEEP within 2 hours of waking, so start getting him ready for his nap after he has been up for 1.5 hours. You have to find the sleep window.
He keeps waking because he can't self-sooth, so after each sleep cycle, he wakes up. The nature of baby sleep changes at 12 weeks and becomes more "adult like," so babies don't just fall asleep anywhere, etc. they have to learn how to sleep. Make sure you are putting him to sleep AWAKE. When he pops awake after being put down, start over, nurse until he is relaxed and keep trying to put him down until he finally puts himself to sleep. It takes time and a lot of work, but in a few weeks, he should have the skills and he should be sleeping better.