SRAM brand MRX 3x8 will work for you.
they come with fresh shift cables installed at the factory.
search: SRAM MRX 3x8.

as to the cable housings, i'd strongly advise building/using fresh ones now, instead of having trouble later.
re-using cables and housing is a not a good choice, especially when the shifters are having trouble already, eh?
plus, re-inserting frayed cables is usually a mission to total frustration.

i still ride my DB Ascent from time to time.

this might be the perfect time to step up to Trigger shifters, btw...
Search on ebay: Shimano Altus SL-M315 Rapidfire+ 2x7/ 2x8/ 3x7 / 3x8 Speed Trigger Shifter
then select 3x8 and front & rear.. under $30, with cables already installed.