K.,
I have a 4mo little girl who didn't nap well at 10 weeks either. My breastfeeding didn't work out and we had to put her on formula at 5 weeks and it took several weeks to find one she could tolerate and she had LOTS of gas during that time. I thought that she wasn't napping well because I held her too much during her colicky phase and/or because she was still having gas and/or because I was failing at being a good mom (I know how you feel!). Plus, I was sooo tired, I just wanted her to get some rest so that I could, too.
Around 8 weeks old (she was still very colicky) I had started laying her on my chest on her her stomach (it was the only position that seemed to bring her relief) and letting her nap there for a while. She would still only nap about 30-45 min at a time, but at least she was getting some sleep
I worried about coddling her and my husband told me he thought I was spoiling her, but I swaddled and held and rocked and sang anyway.
The good news is that she finally started napping better at about 11 weeks (you're almost there!), but she still needed to be swaddled for a good nap then. I didn't hear about the 8 hour rule, so I still swaddled anytime she seemed to need it (her little arms would just flail and wake her up if she wasn't tightly swaddled).
At about 11.5 or 12 weeks, she started sleeping through the night (about 10 hours at a stretch) - tightly swaddled.
She's now 17 weeks and she naps during the day without being swaddled (sometimes in the swing). We still swaddle her at night, but not as snugly and sometimes with an arm out.
I would rock her during the day and let her fall asleep in my arms (even though they say you shouldn't) because it was the only way I could get her to go to sleep. And, I can put her down now while she's still awake and she will fall asleep. I had to slowly transition her to that over the past couple of weeks, but it went fine... I just kept putting her down sooner and sooner after picking her up to rock her until I would pick her up, put in my arms as if to rock her and then gently lay her down.
I know how tired you are now and how hard this is, but it does get better! I agree with the other mom that said the most important thing is that she get some sleep right now - not how she gets it. You can work with her on the how later.
Hope that helps. Hang in there!
P.