I have/had both varicose veins and spider veins. Yep, both can be made worse by excess weight, though normal weight people get them too.
Spider veins are easily treated with injections. Varicose veins can sometimes be treated with injections and sometimes surgery is required. I believe there are some newer laser therapies that work for some varicose vein cases. It's worth checking with your doctor.
I had my varicose veins surgically removed in both legs after weight loss. Unfortunately, they were more noticeable after I lost weight -- I guess they came out of hiding in all that fat!
Insurance paid for the surgery because a Doppler test showed that I have something called chronic venous insufficiency in both legs. It basically means that the valves controlling the blood flow aren't working properly so my legs swell. I believe that my many years of obesity cause the underlying vein problems.
I now wear (expensive!) compression pantyhose, especially when exercising, and am hoping that will prevent any further varicose veins from forming. Something like that might help yours from getting worse.
So far as my experiences go and based on all that I've read, weight loss and/or exercise alone won't make spider or varicose veins go away. Once the damage is done, it's done, I'm sorry to say.