Inspiron 3847, fans won't stop revving, #3

Greetings to my fellow Dell users:

For some time now--several months, to be exact--the CPU fan in my machine has been "revving" nonstop, from the time I power on the computer until I shut it down.  At first, the revving was gradual-only when the machine became overheated (more or less normal).  But something occurred within the fan itself which caused it to breakdown in this fashion.  The "revving" won't stop, even when there is no need for it to continue (I measure the heat buildup constantly and it stays mostly in the "safe zone").  At startup, before the POST, I receive the error message "CPU fan failure," and press the F1 key to continue as instructed.  But any work on my machine is restricted for obvious reasons--the noise from the "revving" proves too annoying to work for more than a few minutes at a time.

I've been seeking replacement fans on the internet that match the stats required for installation on my computer model.  One fan seemed promising, and I ordered it (from eBay; my computer is going on ten years old and new parts for it no longer exist).  However, when I received it, it looked to be the same in every respect except one, which I noticed right away.  The heat sink on the item I purchased measures one-and-a-quarter inch deep, whereas the heat sink on the fan that came with my machine measures three-quarter inches.  As you all know, the breadth of the heat sink makes a difference, even if it can be used on the same model, although I am not sure what that difference is.  I am hoping that I may encounter some expert advice on whether or not using it would be wise. 

Purchasing an item on eBay, particularly for a computer that I purchased in 2014, is an act of faith; obviously it is used and there is no way for me to know how or for how long it was used.  But I have few other choices.

This is a long post, but I am unable to purchase another machine at the drop of a hat (ha!  I am not alone).  So, any workable advice on how I can extend the life of my machine will be greatly appreciated, by both me and my computer.

CartoonMan