The connection string found in <appSettings> section is called the "legacy" connection string and is still used only by a couple of "core" modules such as Documents and probably a few older third party modules.

Are you able to identify what module's data is being read from or written to the old site's database? If the module supports export/import that may be your best (all be it not perfect) way to transfer the data to the new site's database.

What version of DotNetNuke was installed on the old and new sites?

As far as updating the legacy connection string in the web.config for the new site, it's value attribute should be set equal to that of the connectionString attribute of the primary connection string found in the <connectionStrings> section of web.config.

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Bill, WESNet Designs
Team Lead - DotNetNuke Gallery Module Project (Not Actively Being Developed)
Extensions Forge Projects . . .
Current: UserExport, ContentDeJour, ePrayer, DNN NewsTicker, By Invitation
Coming Soon: FRBO-For Rent By Owner