I'm about to check if the paste is still up there
What do you mean by "
if the paste is still up there"? Where are you thinking it might go?
For sure, when you mounted this new CPU, you should have applied a fresh new layer of TIM (thermal interface material). If you used the same old heatsink, you should have completely removed the old TIM and thoroughly cleaned the mating surfaces before adding any new TIM. Did you do that? Was the new TIM a pad, or paste you had to spread around? If a paste, did you use as small an amount as possible while still ensuring thorough coverage? If so, then there is no reason to worry about the paste doing its job or if it "
is still up there"? Some excess might be squeezed out when you clamped the cooler in place and that's okay as long as it is not a lot being pushed out as that would indicate you put way too much TIM on there in the first place.
Otherwise, the purpose of TIM is to fill the microscopic pits and valleys in the mating surfaces to prevent insulating air from being trapped in there. The best transfer of heat occurs with direct metal to metal contact. So any excess TIM is actually in the way. Hence the reason to make sure any applied layer is as thin as possible. Once properly applied, the TIM will easily last 10, 12, 15
years or longer
AS LONG AS the cured bond between the mating surfaces is not broken. It
never needs to be replaced just because it is X number of years old. Even if it dries out, the solids left behind are still preventing any insulating air from getting in there.
If a CPU "
needs" the few (typically 3 or 4) degrees a fresh new application of TIM
might provide to prevent crossing the excessive heat thresholds, you have other issues that need to be addressed first - like the cooler itself being inadequate, or an improperly cooled case.
What cooler are you using with this new CPU? As seen
here, the X5670 does consume considerably more power and can generate more heat than the
L5640. Assuming you are using a cooler that (when properly mounted) is capable of adequately cooling that CPU, assuming you did indeed thoroughly clean the mating surfaces, assuming you applied a fresh new layer of TIM, and assuming you did properly mount the cooler, I agree with the others and you need to look at case cooling.
I hate to assume and that is a lot of assumptions.

Here's another - I am assuming the case interior is clean of heat trapping dust.
I don't know what "Blue Light Speed Fans" are, how much air they move or how they are configured in your case but it is important to note it is the case's responsibility to provide an adequate supply of cool air flowing through the case. The CPU cooler need only toss the CPU's heat into that air flow. You typically want good "front-to-back" (or bottom-to-top) flow through the case. You need to inspect your case to see if it will support more or larger case fans.
My guess is your "Dell Precision" case does not offers enough cooling in its current configuration, and more importantly, I fear it does not offer any cooling upgrade options. So my guess is you need to shop around for another case and hope and pray Dell did not give you the shaft by going proprietary!