What did you fill the pot with? Hope it was potting mix. If it's soil, that's the problem.

Please post an mage.

"It started to grow leaves immediately, but it has now developed signs of severe overwatering..."

You are vague in what is happening. Yellowing leaves? Too much water.
Potting soil with added vermiculite or pearlite will help drainage. Many plants need to have roots dry out once in a while to help them breath.

What size is the tree?
How old is it?

I believe all plums are deciduous, maybe it is only dropping leaves normally. In a cold winter, all leaves will fall off but in warmer climates, dropping leaves all the time while retaining leaves is normal. I am not sure how plum trees work though.

Sorry to be negative but personally, any company that delivered a bare root fruit tree in leaf to me here in the UK, let alone a city in the middle of a desert, has no idea what they're doing. I'd ask for my money back.

I saw that too but neglected to say anything.
I agree, question the vendor.