I would rather redesign it. As im planning to put a Luhndal LL1941 1:32 step up transformer, to pull my dac to "current mode" which has better perfomance and to make some better tube buffer..
I think that is a good choice. The issues that you have now will be an issue with just about any component being fed by your DAC.
The SUT's are a good idea, I would do that and use a single 5687 per channel. You have 230V of B+, and I would run as much current through the 5687's as you have available (to a point). C4S loading would also be a nice idea, but not completely necessary to start.
Your output impdeance will drop to under 3K with this effort, and without putting pencil to paper, I would expect and order of magnitude less THD at the same time.