More expensive DAC the answer?

Have you talked to GIK Acoustics? They give great advice and have awesome products.

Another vote for a tube amp. I have had Klipsch Heresys since 1984 (now in a second system). For many many years, I powered them with SS amps. Sounded fine, but when i got a tube integrated (Jolida), I experienced a huge jump in sq. I really think those speakers are made for tubes.

Addressing the digital question only (amps & room treatments are worthy side topics, as others noted)--I found a very large increase in my appreciation of digital audio just by ditching delta-sigma DACs and going to multibit, especially NOS (non-oversampling) multibit. After doing so, all that subtle harshness, hyped transients, and shallow 2D note bodies are pretty much gone; music has its true weight & impact restored. The cheapest really good sounding NOS DAC I know of is the Audio GD R2R 11, a combination unit (DAC + headphone amp). You can easily use it as an amp only, however, using USB input/RCA output pair to preamp/amp. You'd get one of the most praised inexpensive NOS DACs around + a bonus (and rather good/strong) headphone amp--for <$300. Just sayin'...

correction: "You can easily use is as a DAC only, however..."

While upgrading your source will probably give you an improvement, and going to tubes will most likely sweeten up the sound, it sounds to me that you need to work on speaker placement.  Klipsch speakers can get a little shouty, especially when paired with a brighter integrated like the Yamaha.   Good speaker setup is crucial.