As you state queries are taking longer the longer the server is up, would you be able to enable profile timings and grab a plan soon after a restart, then again (for the same query) later when it has begun to execute slower - perhaps operator stats will give a hint as to where the delay is building up.
OK, since you don’t have profiling available, perhaps check on process size (e.g. RSS) to see if this is increasing over time. It could be that memory is growing / becoming more fragmented and this could contribute.
You might also simply consider upgrading from 6.5.0 GA to a current release to benefit from the updates we’ve made over the past years. This might short-cut any investigation (as if a known bug we’d have to direct you to upgrade anyway).