The answer to your question really depends on your aldehyde, and how sensitive it is (for instance if it's chiral at the alpha position then you don't want to be doing aqueous work ups with anything basic).
1. Filtration
Filtration is by far the most convenient work up for a DMP as the filtrate is often clean enough to telescope through into another reaction.
Firstly, if your reaction isn't already buffered with sodium bicarbonate, add a spatula tip before trying to filter, it quenches out any acetic acid which will not be trapped by the column.
Secondly, add something like ether or hexanes— the DMP related stuff isn't soluble in these, so they'll just sit as solids on the top of your column compared to if you elute with a tonne of dichloromethane for example.
As you've found, it can be a little difficult as the amount of solid gets quite large and traps product. Using a sintered funnel under vacuum makes it far easier, as does using a wide but thin plug rather than a long column.
2. Aqueous workup
I often avoid this one because if I'm doing a DMP it usually means I'm making a very sensitive aldehyde and don't wish to have it sitting around removing the last traces of water afterward.
As AirHuff notes, aqueous thiosulfate washes are very effective. As previously, I'd suggest initially quenching with sodium bicarbonate (a solution this time), then extracting with ether, followed by a final thiosulfate wash, drying, and evaporation.
3. Avoid DMP entirely
As a reagent DMP is great for doing small scale reactions as you essentially add a scoop of reagent and solvent in an open flask and it's done in ten minutes.
On a large scale however the procedure generates a lot of solid byproducts, which can, as you've discovered be difficult to remove. There is also a huge cost associated with DMP, even when prepared in house. For the material you have, you'd be better served doing a different oxidation.
