3 things, blades, the razor & the way you shave.
since you switched to new blades they are sharp. You did not drop the razor with the blade in or otherwise dent it. A blade that isn't straight will cut you. How the razor holds it. if its a branded razor, it holds the blade straight, no name ones sometimes don't do that well. If all these are ok then its operator error. You are applying too much pressure or you're not getting the
blade angle right. For this you have to develop a feel of how the blade cuts, how it feels. Idea is to let it do the work and just guide it. I'm assuming you throw your head back to stretch the skin when doing the lower neck which is more sensitive than the face. Unlike straight shaving you don't have to stretch the skin much with DE just have a level playing field with no skin folds in the way. Give it a rest for three days.
WS that were made in England over thirty years ago were very good. There are supposed to be two types sold in India, one going for around the Rs.30 mark and another for Rs.12. Never found the expensive ones and have tried the cheaper. They were ok, nothing great.