Networking – Why do the speeds fluctuate from decent to unusable while on wireless

I'm getting roughly 116 Mbps down

This would imply that this is a local gigabit connection between the cable modem and PC.
The WRT160N only has 10/100 Ethernet ports, whereas the Cisco modem does have a gigabit port.
You would need to upgrade to a router with gigabit WAN & LAN ports in order to preserve this downstream rate.

Here's a chart indicating the range of WAN-to-LAN-port performance among various routers. If the measurements are accurate, then even "gigabit" ports are not a guarantee for fast throughput.

ADDENDUM

Because how else could they advertise 300 Mbps wireless speed? Thinking about it again now, I suppose it could be shady advertising with the 300 being for ad hoc network purposes only.

That's not shady advertising at all.
The wireless link is fully buffered from the wired links, so the wireless speed is completely independent of any wired link speed. Each link can transmit & receive at its full speed without regard to the speed of any other link.

Almost all low-cost Ethernet switches employ the store-and-forward (rather than cut-through) method, which also means that each Ethernet frame has to be completely received before transmission can commence to the destination. Routers and switched are computers that have local RAM to buffer/store the Ethernet frames as they are received.

The widespread "weakest link in the chain" analogy applied to throughput is often misinterpreted as if all links get slowed down to the speed of the slowest link of the communication path.
That is not a correct view of the data transfer.
Digital data is almost always transferred in packets or datagrams or frames (rather than streamed as continuous bytes) and these packets/datagrams/frames are almost always fully buffered when received and then forwarded.
So there is no dependency of the speed on one link with another link.

A low average throughput number does not mean that a high-speed link was actually slowed-down by another link in the path.
It's the dead or idle time of the transmission links (and the the processing time spent by the router or switch) that gets averaged into the total time to transmit that causes a lower average throughput number.

The specs for the router state 802.11n is capable "Up to 300Mbps(dynamic)". That doesn't mean it will operate at that speed all the time. Theoretically, two wireless devices may be able to transfer data over the local network faster than wired devices limited to 100Mbps.

Network speed is limited to device capability. The slowest device in the connection path (modem, router, network card) will always be the limiting factor. Your router may be capable of 300Mbps with 802.11n, but an 802.11g device will only be able to connect at a maximum of 54Mbps.

Lastly, whatever your modem's capability is will determine how fast your Internet connection is. You can spend a lot of money on a fast router, but your Internet connection will only be as fast as the modem is capable.