My period started on the first day of Ramadan, but when I woke up the next morning I found nothing, so I took a shower and made my intention to fast for the rest of the day. Then after Dhuhr prayer, I found a trace or two of what I think was menstrual blood, so I immediately took another shower and continued my fast. Now, I am worried about whether my fast was valid.

Your fast on that day was vitiated due to the discharge of menstrual blood. Fasting is deemed invalid in this case even if blood comes out a short time before Maghrib.
And God Almighty knows best.