Is what I did, by not repeating the prayer after finding a red or brown spot on the tissue following my Asr prayer, performed with the ablution of the Dhuhr prayer, correct, or must I repeat it?

The ritual impurity (hadath) is estimated to have occurred at the nearest possible time. If it was not likely to have occurred before, during, or after the prayer, then it is estimated to have occurred after it. Based on this, your prayer is valid, and you are not required to repeat it, for "the origin of every occurrence is to estimate it at the nearest time." It is permissible to perform more than one obligatory prayer with a single ablution (wudu') as long as ritual purity is not invalidated, even though it is preferable to renew ablution for each prayer, even if one is already in a state of purity.