The mistakes come in P(1 empty urn) and P(2 empty urns).

The mistake here is this: you are assuming that it is not until the fourth ball that one of the urns gets a second ball. But there are other possibilities. For example, the first two balls could go into the same urn, and then the third and fourth balls into empty urns.

One way to calculate this probability is as follows. There are 4 ways of choosing which urn is to remain empty. For each of these, there are 3 ways of choosing which urn gets two balls, and ways of choosing which two of the four balls go into that urn. Finally there are 2 ways of allocating the remaining two balls to the remaining two urns. Total: 4×3×6×2=144 out of 256.

Same sort of problem here: you could put the second ball into the same urn as the first one, then the other two balls into another urn. The total here should be 84 out of 256. Then the probablilities add up to 1.