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It is likely that you are experiencing the symptom of episcleritis or scleritis due to underlying autoimmune
connective tissue or collagen vascular condition. This symptom of eye vein/capillaries prominence can be seen with chronic use of visine drops to clear redness from eye allergy. So, this could contribute to your problem. I suggest that you follow-up with your eye doctor for closer complete eye evaluation with tests including ANA, ESR, C-reactive protein, TFTs, and ANCA panels. In the meantime, I suggest that you use over the counter Zaditor eye drop 4 times daily, and preservative free tear drop 4 times daily, at least 10 mins apart from the use of Zaditor eye drop. Also, use over the counter Aleve 1-2 pills twice daily with food for 2-3 weeks. In addition, I suggest that you stop the use of visine. Additional treatment or changes, if any, will occur after you see your eye doctor.
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Please note that this answer is for information only. It cannot be substituted for the visit to or the recommendation of your doctor.