I suggest starting with a regularly scheduled profiler trace, capturing Batch and RPC Completion events and certainly the database amongst the columns.

You could have it scheduled for several times a day, running for say 5 mins and a maximum of 1mb output file. The output file name should be constructed from the current date/time.

If the output is too "noisy" you could filter out very low cpu and/or duration events.

Having this information available allows you to load it to SQL tables or query the trace files directly from QA, and thus identify trends and possibly the culprit.

Speaking of trends... does job history show the execution times of the new database's scheduled jobs starting to blow out at some stage?