I don't know the Dell Laptop in question, but fans do get clogged with dust and fluff, and stop working.
That will mean the chip isn't getting cooled, so will overheat and eventually shut down. The stalled fan will also draw more current, and provide more load on the power supply, which will overheat.
Cleaning the fan is possible, but in my experience a clogged fan is usually beyond redemption and is best replaced. Dismantling a laptop isn't for the faint hearted, and I would only do it if I had the instructions on how to dismantle (and reassemble!) it.
I have the complete PDF of the service manual for my (now vintage) IBM Thinkpad, and have had the fan out of that (to apply the thermal compound between the chip and the heatsink that was omitted in manufacture).
A typical computer shop won't have a clue about how to fix it. If you can find a local workshop-type chap who knows, you may be able to get your fan replaced for a reasonable price. Otherwise, new laptops are a lot cheaper these days
Which brand to buy is a different issue...