Grumpy_Mike: Those FETs require 10V to switch on the gates. Don't be fooled by the gate threshold voltage that is just where it starts to cinduct. I am not sure if that would cause your reported problem because it seems they are on all the time rather than not on.
I'm glad you mentioned that as the data sheet did confuse me quite a bit. The gate threshold voltage shows a min-max of -2.0 to -4V and the following was in the comments on the product page:
The product description also says: The FQP27P06 is a very common MOSFET with very low on-resistance and a control voltage (aka gate voltage) that is compatible with most 5V microcontroller or mechanical switches. This allows you to control high-power devices with very low-power control mechanisms.
But again, I'm so ignorant with this stuff. Earlier you indicated that this looks like it would work, but was just over spec. But is it now looking like it won't work at all without 10v and the product page just has some mis-information?