Should be a Hardware Monitor section in BIOS with CPU temps.
Thermal compound --> remove HSF, clean compound from heat sink and CPU using isopropyl alcohol, apply new thermal compound and reinstall. Use manufacturer's instructions for applying thermal compound - a thin layer is sufficient. If you are not familiar with removing and installing an Intel HSF, post back.
Was old stock cooler rated for E6300? Most Intel Socket 775 coolers should be adequate.
Check 4 HSF retainers before doing anything else.