i would double check your connections first. if it starts up once in a while and you hear it click, the relay should be fine. clean the connections, and make sure they are secured tight and you are not grounding the starter to a painted surface. let us know what you find.
if what i mentioned above doesn't work, try removing and cleaning out your auxillary air valve with carb cleaner and a rag or q-tip. the AAV is the trumpet shaped thing that is bolted onto the intake manifold right behind the throttle body. the AAV is really sensitive to little crap and can gum up.
try cleaning the electrical connection to your cold start valve. had the exact same cold start issue. this fixed it for me... also, make sure the connectors to thermotime switch and watertemp sensor, which are both located at the thermostat housing, are not switched around. this fixed the hard warm start issue i was having. good luck, hope this helps...