heres what you gotta do!
get the seed on your survival map,
make a new world with that seed in CREATIVE mode,
and just fly around in the direction you think the biome is.
p.s: try to remember landmarks.
p.s.s: you can fly in the direction your facing automatically simply by flying straight and after a couple secs press ALT, your screen will freeze, then take your finger off "W" and click the mouse!
While this won't help you actually pinpoint a mushroom biome, I recommend entering the Nether and picking a direction. Every 1 block traveled in the Nether is equivalent to 8 blocks traveled in the Overworld. This method is obviously a bit more dangerous, but if you're careful it might be a much easier way to find a mushroom biome, and in the very least is an easy way to get very far, very quickly!
Its rare to find a mushroom biome using a seed is a good option or mods and stuff but finding one legit is HARD i found one in a prison server i just went west the hole way and eventually after going around like 3 or 4 island's/ Not island's XD i found a mushroom biome i was like OFMG this was the most legit momment in my life xD so finding a mushroom biome is PURE LUCK
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Im no crazy but know that i think about it that cat talked to me O.o it had wings O_O