It sounds crazy, but an alternative to CC is to put bubs into cot when they are settled, drowsy...but definitely awake. We did this, and our girl learned to put herself to sleep, even if she woke after 20 minutes (she had reflux and I think she would get heartburn). We started at about 4 weeks! We would then kiss her on the forehead, leave the room and listen just outside the door or with the baby monitor. If after 5 minutes she was grizzling or starting to get distressed, we would go in, give her cuddles for about 5 minutes (or until she was settled), then put her back in the cot awake but drowsy.

After 3 or 4 days, she started to sleep for longer (rather than 20 minute cat naps) and at 8 weeks started sleeping through the night. She also goes down well for her grandma, and is so happy and smiley when she is awake due to her quality of sleep, I'm convinced we did the right thing.

If she ever started crying or became distressed (like an increase in pitch in her voice), we always went to her). We didn't want her to think that we would not be there for her if she ever needed us...

All babies are different too though, and I think even a baby 1 week or 8 weeks old has a distinct personality, and you should consider that too.

Hope this helps!