Should I owe capital gains on a primary residence sale if I purchased another primary residence in the same calendar year?

That depends on whether you made a profit or not. The two transactions are not related at all and haven't been since 1997.
Form 2119 was discontinued by the Taxpayer Relief Act of 1997. Since that time, you can not defer capital gains on a home by buying one of equal or greater value. What you can do, if you owned and lived in the home for 2 of the 5 years ending on the sale date, is to not have to pay tax on a gain of up to $250,000 (Single), or $500,000 (Married Filing Jointly). Note that if you ever used the home as a rental, then the exclusion does not apply to any gain equal to the depreciation that you claimed or could have claimed.
For further info, see IRS Pubs, 530, 523, and 17
http://www.irs.gov/pub/irs-pdf/p530.pdf
http://www.irs.gov/pub/irs-pdf/p523.pdf
http://www.irs.gov/pub/irs-pdf/p17.pdf

Thanks Jerry, I did live in the home for 5 years, and the gain was under $20K, but I am not sure how to take the exclusion in turbotax.

Just go through the Sale of Home interview and Turbotax takes care of it for you.
Type   sale of home   into the search box, click Find, then click Jump to and follow the interview to enter your information.

Thanks Jerry - I appreciate it.
Turns out I had a typo in the date and THAT'S why it was showing up that I had earned capital gains.  🙂

Check out 1031 exchange rules. Under Section 1031 of the United States Internal Revenue Code (26 U.S.C. § 1031), a taxpayer may defer recognition of capital gains and related Federal income tax liability on the exchange of certain types of property.