If fixture has two bulbs then two blacks are hot wires. Two whites are neutral wires.
That would explain the double wires. Also your new fixture goes white to the two whites, then the black from the fixture to one or the other or both blacks. I would try hooking it up to the whites and one of the blacks, then try the light switch.
If it works, you're mostly there, you just have to figure out where that other black wire goes, probably to another light Fixture? If the first black you chose doesn't turn on the light, try the other one.
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