OK, back from the dr.
She thinks my veins are absolutely normal, and it's common for them to get thicker when there's eye strain. She looked inside the eye, on the retina etc. she thinks I'm all good and I should use the computer less often, with breaks. I think I asked her like 5 times if a tumor behind the eye could do that lol but she kept reassuring me.
At least I can have some peace of mind now... it was worth it.
So, for anyone who ever has this symptom and googles my post - this is probably eye strain & it's normal for veins to get thicker when the eye muscles are overused.