Assuming these problems where not present before the application of the thermal compound, they could be attributed to that.
I suggesting you power off the Pi, carefully remove the heatsinks and gently wipe the board down with some 99% IPA (Rubbing Alcohol). Be sure to use something like a microfibre cloth or a coffee filter.
Leave your Pi for 24 hours, then power it up and see if the issue is still present.
As an aside thermal throttling isn't a huge issue with Pi's especially the Pi 3 due to its superior CPU performance over previous models. The Pi will only thermal throttle at 85°C. The only way you will get it that hot is if the CPU is continuously running at near 100% performance for very sustained periods of time. (In which case the Pi might be under-powered for your given application). Or you are running it in a very hot environment.
What most people forget about heatsinks is they are only really effective if there is constant flow of air over them. (I.e from a fan). I don't think a tiny heatsink on its own without a fan is going to make a huge difference to temperatures.
I would advise once you have your Pi working again to try using it without the heatsinks and see if you notice any difference. If you insist on the heatsinks, use an incredibly small amount of thermal paste. I'm talking a really small drop, the SoC is really small, especially compared to a desktop CPU. As an extra precaution you could use none electrically conductive thermal paste. So, if it over runs it won't immediately cause any shorts. And be sure to remove the excess with some Rubbing Alcohol.