Of course this bothers you! What mom, even a supermom, relishes the idea of having a child clinging to them all afternoon? I know exactly what you are talking about and don't feel badly about being bothered or feeling impatient-- it's human and you are being stretched.
It sounds like your little one is tired too. I agree with Susan, start the nap earlier. Two is fine. I nannied little ones for years and discovered that when I was more receptive to getting them a bit before they were overtired naps often went better because the child themselves has to participate in the routine of going to sleep, even if it 'just' means getting to sleep. (I know it might not look as if she's overtired, but overtired can also look like 'excited' and 'active' even 5 minutes before they crash/meltdown....we know it because of the signals you've described-the "babbling/fussing/crying/babbling, etc etc"... that occur. If she wasn't overtired, she wouldn't get so upset; she'd just be happily playing.)
One thing to try would be to shorten the morning nap to one hour, or try that routine earlier. Optimally, though, you want her to be rested enough to be awake later, so that she sleeps later. If it were me, I'd start troubleshooting by leaving the morning nap as-is and introducing the second naptime at 2 or 2:15. Especially if children are in a growing phase, they need lots of sleep. If that didn't work after a week, I'd try shortening the morning nap and keep the 2ish afternoon nap. You could even shorten the morning nap by an hour just to try things out, and see what happens--fine tune it from there.
It sounds like you are very connected with your little girl. Keep working with her and as Lisa mentioned, this is a challenging time in general, so don't get too discouraged with yourself. It's a dance--this rhythm between ourselves and our kids, and they are at best 'grossly' communicative. Best wishes!