I've run Swerns with a significant excess (2-3 fold) of DMSO compared with (COCl)2 before and they worked fine, so I doubt using DMSO as a cosolvent will be a problem.
A practical tip though: Adding a DMSO solution to a vessel at -78 °C tends to end up with a needle blocked with frozen DMSO, which is a total hassle. I recommend adding your alcohol in a DMSO/DCM mixture - the DCM is basically an antifreeze here, 20% DCM content should be enough to keep your needle clear during the addition.
THF can also be used for Swern oxidations. By analogy, I guess (but have not checked the lit.) that you could also use 1,4-dioxane or 1,2-dimethoxyethane. If any of those solvents will dissolve your starting material, they might be worth considering.