Hey kdawson,
Congrats on your new LO! I can only share my experiences with early sleep training...
I started trying to figure it out at 4 weeks, but we didn't truly implement it correctly until 6 weeks. She is now 9 weeks. Honestly, I feel like we pushed her too far too fast. I learned that the important thing is to follow their lead. If she constantly fights being put down awake right now, I would hold her a little longer. Take baby steps for several days. When we pushed too far, we wound up with other issues such as waking up after 20-45 mins constantly and her crying herself to sleep. Therefore, I'd take it easy since she's only 5 weeks old. That being said, you need to take care of you! So stick her in between two pillows or rock n play when she sleeps to feel snuggled. Or place her by your leg for a nap. That way she can feel and smell you, but you'll have your hands free. Honestly at this age, its not going to ruin her.
Do you lay her on your chest with skin to skin or clothes on? Also does she sleep on your chest on his stomach and sleep in the crib on her back?
When she wakes up and you do the routine, Tracy says when crying we're suppose to pick them up and do the shush/pat. (The shush pat drives my LO crazy.) But do pick her up and calm her down. Try to have as little stimulation as possible. For my little one, i can't look her in the eyes or move very much. Every child is different though. And if you do the movement (shush pat, rock, anything) only do it to calm them back down. Then stop the movement for awhile and hold them. Then a few minutes later, depending on their comfortability with being placed in the crib, put them down.
I read one of Tracy's interview where she addressed short naps. I'm not sure if this is the case with your child since she so small, but Tracys said her child was overtired and not eating enough. Therefore, she wasn't filling up enough to last the entire nap. Therefore, it was a vicious cycle. This was the case with us, so we changed her schedule to last longer between feedings. And we started by trying to get her to sleep longer so that she would eat better. She still falls sleep briefly in feedings, but I think that simply in response to the "Sleepy hormone." (Forgot the name of it.) Plus at that age, sucking for food is hard work. So I'd suggest working on getting her to sleep longer, and the feeding might take care of itself.
Good luck! It sounds like your doing a great job!