Over heating

I have a 2008 Chevy express van 5.3 vortex engine.
About this time last year I replaced the water pump
and put a new thermostat in twice. Well it started
running hot again two weeks ago. I replaced the
thermostat and it did ok for a few days and then
started over heating again. This time I checked the
water and it was low I added water and it seemed
to be fine. Drove it over the next few days no
problem. Now it's overheating again. I took the
thermostat completely out this time and trying it
without one. I drove it down the road a couple
miles and the computer screen says engine. Only
this time it's not boiling over and It didn't reduce
power like it did before. So my ? Is...is it OK to run
it like this and is it saying engine hot because it's
not registering because of the thermostat being
out or what? I realize the heater will not work like
normal with it out. But I'm just making sure it's OK
and I'm not about to ruin my motor. Thanks for any
help.