Progress on this puzzle seems to have two parts.
In the first stage of solving, we can determine a little bit about the blocks by comparing pairs of words.
(1): Comparing BOXY and OKAY, we note the matching of O and Y, and we see that either one block must have AB and the other KX, or one block has AX and the other BK.
However, BK would not be possible, because then we couldn't spell BUCK. Therefore, we know that one block has AB, another has KX. We also know that the remaining two blocks have a U and and C.
1: AB****
2: KX****
3: U*****
4: C*****
(2): Consider EXAM and OKAY. These two words require us to assign Y, E, O and M to blocks 3 and 4. Moreover, the pairs are either (YE) and (OM) or else (EO) and (MY). But if E and O were together, we couldn't spell GORE. Therefore, the split must be (YE) on one block and (OM) on the other. Because we can spell JUMP, the M can't be on block 3, so (OM) goes to block 4, and (YE) to block 3:
1: AB****
2: KX****
3: UEY***
4: COM***
(3): Because we can spell JUMP, P and J must be split between blocks and 2. But in order to spell PAWN the P can't be with A.
1: ABJ***
2: KXP***
3: UEY***
4: COM***
(4): What do we do with the W and N of PAWN, which must be split between blocks 3 and 4? Because we can spell CHAW, the W can't go to block 4. And that also means that the H has to go with block 2.
1: ABJ***
2: KXPH**
3: UEYW**
4: COMN**
The next few deductions become a little harder to work out.
(5): From GORE and GIRL we can see that G and R must be split between blocks 1 and 2, while I and L split between blocks 3 and 4. But that means that the F and T of FLIT must also be split between blocks 1 and 2. So at this point, the two empty slots in block 2 are filled, although we don't know exactly which letters go there. And similarly, two of the free positions in block 1 are now actually filled, although we don't know exactly how, leaving one completely free position in block 1. We'll let "?" indicate that we've reserved space for the letters F/T and G/R in blocks 1 and 2.
1: ABJ??*
2: KXPH??
3: UEYW**
4: COMN**
Knowing that block 2 is filled, consider the information from ZEST and DIGS. We already know that T is in block 1 or 2, and we know that E is in block 4, so one of Z and S goes to block 1, and one to block 4. Meanwhile, we know that the D and S in DIGS must also be divided, one going to block 1, and the other to blocks 3 or 4.
In other words, either S is sent to 1, while D and Z go to 3 or 4, or else D and Z go to block 1, and S goes to block 4. But there's only room for one more letter in block 1. Therefore, we can't cram D and Z there, and instead S must go to block 1. If Z has to go to blocks 3 or 4, it must actually go to 4, since it can't be on the same block as E.
1: ABJ??S
2: KXPH??
3: UEYW**
4: COMNZ*
(6): To spell ZEST we'll need to put the T in block 2. To spell DIGS, we know that G and I are split between blocks 1 and 2, and we need a G, and the G can't go in block 1 because the S is there, so G must go in block 2, and that puts R in block 1.
1: ABJR?S
2: KXPHTG
3: UEYW**
4: COMNZ*
(7): We already know that L and I are split between blocks 3 and 4, but in order to spell DIGS we still need to place the D in block 3 or 4. But there's only one space in block 4, which is either L or I, so D must go in block 3.
1: ABJR?S
2: KXPHTG
3: UEYWD*
4: COMNZ*
(8): Spelling FLIT means the F goes in block 1.
1: ABJRFS
2: KXPHTG
3: UEYWD*
4: COMNZ*
(9): Spelling DIGS requires the I to go to block 4, and that puts L in block 3, and we've placed all the letters!
And thus our final arrangement must be:
1: ABJRFS
2: KXPHGT
3: UEYWDL
4: COMNZI