It seems your PC is a AIO type. These are nothing more than slim desktops that use laptop type components, etc..
One area that make these suffer is "heat stress" and that generally refers to a weak mtrbd. or sub-PCB found on the mtrbd. It maybe a device found within the case, but generally they're OK. heat stress simply makes the overall stability of the PC get as you've experienced NOT always reliable on bot or prolonged usage. I recall trying to help a similar PC from Asus they provide PC for other OEMS like Gateway, HP, etc.., but he couldn't reaily find a replacement part. Point blank if you have any warranty, use it. If you can't seem to fix this problem, then more than likely its mtrbd. based, so research the cost aspect of this or repair. Closely inspect any USB connections or hubs int. the PC for bad lands(think blown fuse).
As for find 18.7V on the 19V AC adapter, that's OK. Generally, anything within 5-10%. You may want to reduce the load and see if that helps, disconnect to make a simpler PC. Also, I use a magnifying glass to better inspect the innards, it helps. Also a "bad cap" needn't be so obvious, a slight bulge maybe all there is or "discoloring" directly beneath the cap(leakage).