Baby sleep problems!! PLEASE HELP!!

Koda
My daughter is 3 1/2 months old. She'll be 4 months on January 29th. 
We got a swing for Christmas. Before then, she absolutely refused to sleep all day long, until she was over tired all the time and crying and crying. I had to swaddle her and walk around the room rocking her for 30-40 minutes every time she needed to sleep, which killed my back.
Now that we have a swing, I've made it my job to make sure she gets enough sleep. We're also putting an emphasis on keeping her on a schedule, which mostly means she wakes up in the morning a certain way, and goes to bed at night a certain way. The rest of the day is mostly unscheduled. 
This is what happens when I put her down for a nap:
I make sure she has lots of tummy time, playing with me and with toys, feed her until I'm sure she's full, burp her, and of course make sure her diaper is clean. 
I put her in her swing an hour to an hour and a half after she woke up from her last nap. 
I set the swing to the fastest setting, give her some quiet (and not soft) toys to play with to keep her distracted and calm, turn on the music on the swing, and walk across the room back to my bed to do whatever I normally do.
The second I put her down in the swing, she starts crying. I let her cry for a few minutes and if she cries too hard I go over to the swing, switch the swing so it's rocking her from head to toe instead of side to side, get her interested in her toys again, sing her a song and talk to her, and she's calm. 
Then the second I walk away again she's crying just as hard as she was before. I wait it out for a minute or two and try again.
Earlier today I decided to try letting her cry for 5 minutes. Before those 5 minutes were up, she started screaming at the top of her lungs, then suddenly "crashed" and just fell asleep in the middle of crying.
I've tried that a few times today and every time I do it, it takes a minute or two longer for her to fall asleep, but she does eventually.
What is your advice for me?