I really don't think heatsinks will cut it, 3+1 phase power really isn't built for 125W processors. I knew a couple of people on ASUS 3+1 boards using 125W processors on them; the OCP (VRM anti-fail) activated way too often at load, throttling the processor and reducing performance, because the VRMs overheated and overloaded. This was with these CPUs at stock speed, on the stock cooler (so VRMs are even getting some air cooling).
New Gigabyte boards with the exact same VRM design are not being rated 125W ready. Their lower-end 3+1 design with cheap 3-transistor fets and also does not feature OCP, so they are actually known to fail, nearly as much as most MSI 3+1/4+1 boards. This is why you really must invest in a new board! Or begin undervolting (underclocking?) if you wish to remain on the current board.