Hi C., I see you've had a lot of responses already but I have to stop to respond because my baby just turned 3 months old 2 days ago and is the same way. She nurses less often--every two hours or so during the day--but I can't put her down or she'll cry, and she hates the swing. And I'm not freaking out, but the ONLY reason is because this is my second baby and my first was the same way at this age, so now I know... this is not forever, this is not something I need to change. This is something that will change on its own, sooner than I know. My new-mama mantra is "this too shall pass."
It's easy to feel trapped, like you need to "fix" your baby, and I see a lot of well-meaning advice here, but honestly, it doesn't sound to me like there is a problem. As long as her growth is staying constant, your milk supply is probably fine. You may just need to find a way to get your hands free just enough to keep yourself from going crazy. Your doctor is right: In another month or three you will have much more freedom than you do now, so you just need to cope for now. I'm not sure what kind of carrier you are using, and I see others recommending various kinds, so I'll tell you that I mostly use a Moby right now. It will probably only be the best for another 2-3 months because it's good for small babies, but my baby LOVES it and it's very comfortable. It holds the baby close to your body, which babies like mine (and yours) often love, is very secure, and keeps her out of the way. When my babe is fairly calm, I pop her in the Moby and she falls asleep within a minute or two. Yes, she's "in the way" in that I can't bump up against the kitchen counter or whatever, but it keeps her asleep and I can do things with both hands, and that is a sanity saver for the moment. So I do recommend a Moby for new moms if it works for you. Or try other carriers if the one you have isn't working for you. I have a million (OK, like 6) and different things have worked for each of my 2 babies. A Hotsling is also worth trying; it's very easy to use and a lot of babies and mamas like it. A trick with the sling is to start walking the second you put the baby in it. Even if she fusses for a second, the motion of walking may lull her to sleep quickly.
My baby is also gnawing on her hands and slobbering and crying. I *KNOW* she is teething, even though her gums aren't bumpy yet, so I know I'm in for months of this probably. And it seems to really bother her. Once I gave her ibuprofen and it helped a LOT, so if your little one seems miserable and won't stop crying from teething pain, try that instead of acetaminophen. But otherwise it's something you both have to ride out together. Pretty soon she'll be able to hold things in her hands and she can chew on teething toys and stuff. Again, this too shall pass. I know it's hard to believe, but this time will seem like a blip when you look back on it. You just have to give yourself enough relief that you can cope. Breathe, chant "this too shall pass," and find comfort in the little things that work for a few minutes... or a few seconds!
I was never much for changing what I eat or drink because of breastfeeding. If you look at the science you'll see that there are only a few things likely to make a difference in a small minority of babies. If she doesn't have a food allergy don't worry about changing your diet. She's getting a bit old for Happiest Baby On The Block stuff to work, so don't worry about that either if it's not working. Just hang in there, hold your baby as much as you can, give her lots of love and attention, talk to her, sing to her, hand her off when you need to, put her in a carrier and wear her when you can, and keep chanting your new mama mantra. It won't be like this forever. I promise. Hang in there. And feel free to message me anytime. Our babies are within a couple days in age! :)