It does sound an awful lot like Vasospasm. I had it in the early weeks of breastfeeding with both my girls and it's awfully painful. The only thing that really helped me was medication, Nifedipine actually. It's a blood pressure medication which really helps relieve Vasospasm. I also had breast warmers, they offered some extra relief.

Mine was caused both times by nipple damage. Once my nipples were well and truly healed and BFing was going well, it settled. Both times it was several weeks in (lots of feeding issues each time) but the nifedipine really helped while I rode out the storm and then I was able to wean off it completely and didnt experience any more pain for the rest of our BF journeys (11 mths and 2 weeks with DD1 and roughly 1 year and 2-3 weeks with DD2.)

I hope you're able to get some relief soon. It's an awful feeling especially when things like a cool breeze or simply looking in the fridge causes such pain!