I am doing a design for a t shirt so I need to vectorize the image but I did it in photoshop

Hello, you could try to search on the Computer Arts archive if the vectorization tutorial I once wrote is still available, it comes with one vectorization action that you could try...

Will the t-shirt be screenprinted, or it is dye cut?

Please provide a meaningful description and post screenshots to illustrate what you are talking about. 

" messes up the design a little bit" what exctly is messed? Can you share screenshot for better assist?

Why do you need to vectorize your image. I see  Clip art , logos, charts, text  and some  Pattern design are vector graphics.  Images are Raster Pixel images not vector graphics. Images have too much detail .

There are many free T-shirt mockup template on the web. You should download a few and look at how they are constructed  are structured.