1. -느라고 = because I had to do something / as I am/was caught up with something.
2. -노라고 (or -느라고) = (placed before a verb indicating an effort which brought unsatisfactory/doubtful result) "even if I tried my best".
According to the dictionary, only -노라고 is correct for the #2 case, but everyone seems to say -느라고 instead. It appears 느라고 has replaced 노라고 in real life use, so you can probably use either. The two sound very much alike at normal speed speech anyway.
Here's what your examples mean:
- 준비하느라고 했는데 다시 한번 확인해 봐야 할 것 같아요.
= I did my best but I'm afraid I have to check it again to make sure.
- 죄송합니다. 신경을 쓰느라고 썼는데 실수로 빠진 모양이네요
= Sorry, I was being super-careful but it looks like it has slipped through.
-느라고/노라고 is not really synonym to 아무리...아/어도 though. It is said when you have finished something and the result looks fairly reasonable but you have your own doubt or it turns out to be not good enough. like "I did my best, but it might be [has turned out to be] lacking". "이무리 ... -아/어도" on the other hand is used when you fail at something completely no matter how you try, so that there is no result at all. They are not interchangeable.