One thing I've learned with Dell laptops is sometimes the power settings are not optimal. You could try going into your Control Panel, select the Power Options (typically under System and Security - depending on which OS you have). Try using the "Power Saver" profile option.
What that will do is close down a lot of the background processes that are not really needed. As a result, your power consumption is lower, due to less processes running, and that results in a cooler laptop.
If your fan is constantly running, that's probably one thing contributing to the high heat.
FYI - when your laptop shuts down, that's a common failsafe for the hardware, where when it gets too hot, an emergency shut down happens to try to preserve the hardware from frying. Not very friendly for the OS (the hard shutdown), but it's best to keep your laptop from doing that as it's at least annoying if you are in the middle of something.
If that doesn't work, a cooling fan would be my next suggestion. Research online and find one that has fairly high reviews, and it should help.
If it's still running too hot, you may want to take it back in to have the hardware checked. Particularly to see if the thermal paste on your CPU chip is sufficient - that it's fully covered and not dried out. If the thermal paste is not enough for your CPU or if it's dried out, that will cause your machine to overheat. If you are comfortable with doing your own repairs, search out instructions for your laptop model, otherwise take it into a shop.
Hope this helps!