My Sony VAIO laptop had that jet engine sound and was shutting down suddenly on overtemp like your HP. I found instructions on the web on how to get to the fan and heat sink (involved removing the keyboard and other items). There wasn't a dust bunny in there, it was a dust bisquit!!! I'm serious, I pulled it out in one piece with tweezers and it looked like a square of thick felt. No wonder my cpu couldn't breathe. Anyway, since then it has run quiet as a mouse (the fan runs so slowly you can't hear it) and no overheating problems. I only use the laptop when travelling, but I didn't want it to bite the dust just yet. My desktop is a PowerSpec (house brand of Micro Center). I love it because they built it with only the operating system and drivers, no "bloat-ware" to waste space and attract popups that slow things down. The downside is you have to purchase MS Office and other apps separately, but I think it is a worthwhile trade off.
Curtis