Am I having a power supply issue with my LED lights strips in series, or have I done somethine else wrong?
You need to choose a power supply that has the appropriate capacity to drive your LED strips.  Since your strips need 5A at 12V to operate, and you are intending to run two strips off of one power supply, you need a power supply that can supply at least 10A at 12V.

So, search for a 12V power supply that can supply at least 10A.  If you want it to last a while, find one that can supply at least 15A.