Ditto Vegetto.
37C is hot for the case (but not TOO hot), so you might want to check that out as well, but a better heatsink should definitely keep the temps down in safer ranges.
For installing a heatsink:
1. NEVER put much pressure on the heatsink--just the clip.
2. Remove old heatsink (use a screwdriver, should fit into a slot on one side of the clip).
3.Clean CPU core with alcohol or acetone, until the retail HSF compound is no longer there to the naked eye.
4. Put on compound, following thedirections. I'd recommend Arctic Ceramique (Near as good as Arctic Silver III, but doesn't run)...but nothing is wrong with the cheap stuff that will come with the heatsink (assuming you get a Thermalright, as they all have come with some for me).
5. slip on HSF, following directions. And if Thermalright again, remove the plastic film. I've been dumb a few times, don't emulate me (but the mobos did keep it from frying

).
6. If Thermalright SK or SLK series, you can put the fan clips on afterward and the fan if you want. or before, but I find doing it after easier.