Is there a way to turn a normal cow into a mooshroom?
No, they spawn in new mushroom biomes. However, the reverse is possible, using shears:
I also found that my cows had changed.
If you are in a mushroom area, or swamp, and you spawn a cow, it should be normal. If you then leave that cow it will eventually turn into a mooshroom.
It will usually take about 5 hours in real-time.
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