If you are overclocking - stop now! At least until this is resolved.
One of the most common mistakes the less experienced make when applying a new layer of TIM (thermal interface material) is they apply too much! The best heat transfer occurs with metal-to-metal contact. The purpose of the TIM is to ONLY fill (push out the air) the microscopic pits and valleys in the mating surfaces. Any excess TIM is in the way - and actually is counterproductive to the heat transfer process. The second most common mistake is they fail to thoroughly clean the mating surfaces of old TIM before applying new.
So, it is critical to properly apply as thin a layer of TIM as possible, while still ensuring total coverage (I have a canned text on applying TIM I can post, if you want). Also note while there are basic TIM pads and there are high-end pastes, using TIM is MUCH MORE IMPORTANT than which TIM you use. If the extra 3 or 4° you get from an high-end TIM is all that is keeping the CPU from getting "hot" then you have other problems that need to be addressed first.
Is the CPU HSF (heatsink fan) assembly properly mounted? With the temps jumping so high, so quickly, with only a few % points of CPU load, it sounds like the heatsink might not be properly seated, or securely clamped down. Assuming it is good, and the CPU fan spins freely, and of course, the case interior (and vents, heatsinks, fans, cards, boards, etc.) is clean of heat-trapping dust), then it is time to look at the case itself.
The CPU HSF assembly is responsible to "toss" the heat from the CPU up into the "flow" of cool air moving through the case. And it is the case's responsibility to provide that desired front-to-back flow of cool air through the case. I generally recommend at least 2 large (120mm or larger) case fans, (not counting any PSU fans). I like one in front drawing cool air in, and one in back exhausting hot air out (but not all cases support front mounted case fans). Inspect your case interior for fans, and additional (or larger) fan support. If you need to add fans, check your PSU for additional 12V fan (or old style 4-pin molex HD) power connections. Route your cables so their impact on that desired front-to-back air flow is minimal.