I read that it's fine so as long as the motherboard isn't replaced, right?
That's for the operating system, not Office (see *Office exception). A new motherboard is considered a new computer and therefore typically needs a new license for the OS. This applies to OEM/System Builder's licenses which is what typically is used with factory built computers.
You can transfer Office to a different computer as long as any previous installations are uninstalled. And it is really pretty easy because all you need is the Product key or Microsoft account. You can download Office from Microsoft, install it on the new system, then use your product key to activate it. See
this and
this.
*Office exception: If your current Office license is also an OEM license and that copy of Office was preinstalled on the computer at the factory, it is
not legally transferable to a new computer/motherboard.
Also note you can clone your current drive (as long as it still works) and transfer everything to the new drive and not worry about Windows or Office or your other programs. This is commonly done by folks wishing to improve system performance by replacing their aging HD with a new SSD.
Here is a good guide for that.
As far as what you should do, that is always a tricky question - with many things. Do I spend $1000 to fix the engine of my car, or buy a new car? Do I pay to replace the screen on my cell phone or get a new phone? Only you can really answer that.
The problem I see here is your computer is already 5 years old. That puts it right in the gray area of between new and old. Most folks expect their computers to last and serve them for 5 years. After that, things go down hill (or the world pulls too far ahead). The problem is, you can replace your drive, but then what? Have you already replaced your battery? If not, that may be soon. The monitor may start to fail, or something else.
The problem is, you never know. You could be looking at the first of many failures to come, or it may last another 3 - 5 years with no problems. And note as far as buying a new Office license, there are free alternatives like LibreOffice and OpenOffice that are totally compatible with Microsoft Office (except for Outlook).
If me, knowing this notebook is already 5 years old, I would probably start looking at new notebooks. If that was out of the question, I would replace the hard drive with an SSD. SSDs are much faster, lighter, more durable, faster, consume less energy, generate less heat and did I mention they are faster?