Many of the widely available on-line background check services such as Public Data simply report everything in the public record, no matter how old it is. So, the prospective employers may be getting anything in the record. There is no rule that says they can only look back for a certain period of time.
You should ask someone in HR for a finance company this question. I would imaging it depends on the company, I am sorry to inform you that your 1999 conviction will always be found when a criminal background check is done.
Good luck with everything.
Some employers check for convictions; others check for convictions and arrests. Unfortunately for you, the latter group of employers will find about both of your cases. This is perfectly legal as arrests and convictions are part of the public record. As the other attorneys have indicated, criminal convictions and criminal histories purchased from Texas DPS list your entire adult history. In fact, there is no option to purchase only a record that goes 3, 5, 7, or however many years back, as there is with a driving record.
Most employers probably don't care about 1 DWI conviction, but some might think that 2 arrests shows bad judgment on your part, and there's really no recourse for you except for getting your second case dismissed or being found not guilty.
If you haven't already, you need to get a criminal defense attorney with experience in taking DWIs to trial.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.