I need to file 6 returns for our family. I know I can file 5 returns with my desktop version of TurboTax. Can I buy another package and just add the new lic code?

No.   The five e-file limit is an IRS limit--not a TurboTax limit.   You can use the software you already have and prepare the 6th return and file it by mail instead.

Maybe one of the returns can use the Online Free Edition or the IRS free filing options?  So do the most complicated returns first to use the 5 efiles.   And you know it is $25 to efile each state (including the first one).