Soph, the white spots will only appear on the baby, not on Mom. Thrush takes time to develop, according to my midwife and is more likely with cracked and sore nipples. I asked her about it because my nipples felt the same way, but there was no evidence of thrush in Anberlin, and the midwife said it takes time for thrush to develop. It could just be that your nipples are getting used to the "abuse". They'll burn, sting, and have a painful latch until they get used to it, but it doesn't hurt to look into it either.
I breastfed Isaiah for 17 months and went through the same thing, the sore, burning nipples right from the first few days. It wasn't until they cracked at a month old, and a month later, that we got thrush.
If your nipples still feel like they're burning even an hour after nursing, look into a condition called Vasospasm. It's where the blood flow to the nipple will stop temporarily and make it feel like the nipple is burning and sore until the blood flow starts again. I suffer from it too.