First, before you give up... contact a lactation consultant. They are worth their weight in GOLD! I had terrible trouble nursing my DD at first. I didn't have mastitis...but I was engorged all the time, it hurt to even hold my daughter... and we had terrible terrible latch problems. My nipples were constantly bleeding and I was so sore I creid through every nursing session. It was horrible. I was ready to give up. I couldn't take it anymore. I was in so much pain I wasn't enjoying my daughter and even resenting feeding her.
Then I contacted a lactation consultant. It changed everything! She watched how we nursed and worked with us for over an hour to help us 'get it right'. We were doing everything by the book...but the shape my daughters upper palatte just wasn't suited for the 'by the book' method. It caused extra pulling on my nipple etc...
Lactation consultants are trained to know everything about nursing. I'm sure one can help you figure out what is going on.
Also, a friend of mine got a really bad plugged duct. Half of her breast was rock hard! She found some information online that said to nurse on all fours. Basically baby lies on the bed, you hover over on all fours and nurse. Try to massage your breast while you do it. This position helps drain the breast completely... gravity helps.
Honestly, my guess is your body is trying to get used to this supply and demand system. I've had many many many mommies of multiple children say the first time is the hardest... it's like your body doesn't quite know how to 'get in gear'.
It took a GOODO 3.5 months ... but finally DD and I got our rythm down, my supply is just what she takes and I've had no more issues.
Breastfeeding is NOT easy. It takes time to settle into it.
Please... try to exhaust all the options before switching. But, also, if you decide you just can't... then the time you did nurse will be beneficial to your baby too.
Good luck!