Need some help to understand if I have a problem.

Anytime I crankup (hot or cold) and begin to idle it takes 15-20 seconds before I see any water pressure. Normally the pressure is only about 5 psi at idle. If I move forward at a slow speed where the boat is moving at 5-10 miles per hour or with just a little thottle the engine will begin to run a little rough. The water pressure at this point is around 10 psi. If I try to accelerate at full throttle I will get the temp light and horn and of course the engine runs rough as designed. If I shut down the engine and restart (no time delay) I can accelerate at full throttle with no problems. When running at speed (4800 rpms or so) I usually have about 15 psi water pressure. Is this normal also? Is the water temp sensor bad? Could this be a computer problem since I can shut down and restart and take off at full throttle with no temp alarm?

Thanks for any help you can provide.

Charlie