I accidentally cut someone off on the road today. Any help?

Anonymous
So I was driving home and I wanted to switch lanes. The problem was that I had my blinker on and the lady in the lane I wanted to change to was in my blind spot and I saw her in my blind spot. The problem was that the dude in front of me wasn't driving very fast and I didn't want to speed up and hit his car so I kinda stayed back. Then the guy in front of me speeds up and he sped up the lady in the lane I wanted to change to honked her horn at me. I felt bad so I floored my car into the lane I wanted to change to and everything went smoothly from there (I only floored the car to about 40mph so I wasn't speeding). Any tips on how to avoid this the next time? I didn't want to intentionally cut her off but I had to do what I had to do and i don't know how to avoid a situation like that the next time. I don't want to be know as the asshole who cuts people off.
Updates
+1 y
I meant to put intentionally instead of accidentally on the tittle. Sorry
I accidentally cut someone off on the road today. Any help?
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