Power Supply for LED Project
Your calculations are correct. 354 LEDs at 20mA each is 7.08 A (354 * 0.02).

The thing to be careful about with LED strips is the voltage drop. The copper thickness on strips like the 5 meter one you have tends to be 0.5 to 1.0 oz, and the voltage drop is quite noticeable. In other words, if you supply voltage to one end of the strip (12V), the opposite end will only receive 10-11V. The instructions may indicate that you should supply power at both ends of the strip, which is a good idea. Personally, in my projects, I do not run strips longer than 2 meters with that type of copper.

Edit:

Looking at the datasheet for the LED strip, it is a 5 meter strip with 60 LEDs per meter. Each "LED" is actually a 5060 package SMD RGB assembly, which means there are three LEDs per package. The datasheet specifies the maximum If at 30mA (not 20) for each color, R, G, and B. (The characteristics are derived using a test If of 20mA, but the absolute maximum value is a worst-case scenario, which you should use in calculating power supply size.)

Unless I am reading it incorrectly, this means to light the strip at 100% brightness at white (all three colors), it will need 90mA per LED. That works out to 5.4A per meter, or 27A total.

Comments on the product page suggest that the strip only requires 6A. I suggest measuring to be sure. (Strips that I have require about 20mA per color, which is what made me think to come back and add this addendum.)