I'm dealing with a 12 week old as well, and I too came to reading the signals really late. Remember that there is a 12 week developmental shift which makes babies super-fussy at the moment (DS is a bloody nightmare during the night at the moment - we're exhausted). Anyway, I'm doing a kind of Baby Whisperer kind of routine, but it's still baby-led.
I watch the clock after he's been up for an hour and watch for tired signs - he can't manage more than 90 mins - 2 hours awake at a time, which is quite normal. I take action after the first yawn (I really mean it, I whisk him away to his room). I make a big show of closing the curtains, speaking low and soothing, and then I sit with him for 15 minutes or so, quietly. I swaddle him, which often disturbs him again, so I hold him again for a bit. I have a snack available too which can just top him up to go to sleep. I use a dummy where needed. I wait until he's all floppy and dreamy and then I pop him down. Remember not to just take your DD upstairs and dump her in bed - give her time to understand that there is a transition happening and it's about preparing for sleep.
TBH, sometimes it works, other times he wakes up as I put him down - that's a sign that he's overtired, not that he's not ready, as once he's yawning I know he's tired.
DS's tired signs include: stopping making eye contact with me when playing (he turns away), just kind of slowing down and stopping his movements, yawning, pulling/rubbing at his eyes, his eyelids go quite red, he nuzzles into my neck when I hold him, and when he's really tired he kind of makes a creaking kind of cry. That's when I use a dummy - it can be a brilliant indication of tiredness as his eyelids almost immediately start drooping.
Today I have restarted 'nap school' here as his nightime sleeping is so bad that I need to feel that I have some kind of control over something! He managed 3 naps of 40 minutes each, which isn't great, but at least he got some sleep. He's been awake since 3pm now and I spent 90 minutes between 4 and 5.30 trying to get him to have a nap. He was a combination of sleepy, grizzly and beaming with happiness. Little bugger. My back is killing me from picking him up and putting him down. In the end I just ran him an early bath and started bedtime routine.
I've found that even if he doesn't sleep, I sometimes leave him in his basket anyway for a while just so that he knows that there are calm periods in every day where he needs to chill out. I have been known to pick up and shush/pat him for the entire duration of a period where he should have been asleep, only stopping to feed him - it was 4 hours, and he woke up every 6 minutes... It nearly bloody killed me.
Anyway, have a look at the Baby Whisperer - you may find it helps you just to get a sense of some little techniques that could support you in getting some peace during the day!
Sorry, long rambling reply!