I am searching for a quiet 16 port gigabyte POE switch.
It won't be rack mounted.
Seems like, if it has a fan, it's too noisy. Sounds like a waterfall.
I see there are switches out there that use a wall wart for power, and they are fan-less.

I see switches with 16 ports plus a gigabyte uplink to the router.
Do I need that separate uplink port to connect to my router?

Any suggestions?

You do not need the uplink port.
The router can be connected to any port.
The uplink port is usually higher speed and may have some other features you will probably never use than the other ports.
A good idea is download the manual before buying.
You typically need a non-managed POE router.
I like TP-Link.

Something you need to think about is power. If you know how much power each camera requires, add them up and make sure the POE switch has more.

For example, if a camera needs 15W and you have ten of them, you need at least 10x15= 150W but I would go higher to allow for overheating and future expansion.

Make sure you read the fine print. Many of the switches have 16 ports but only 8 of them are POE. Like Tim advises, add up the power requirements.
We have a Netgear 16 port and it is working OK.

What @TimG has stated is very important, make sure there is head room in terms of the power capability.

I purchased an 8 + 2 port Trendnet POE switch and on paper it had enough power for my then 8 Dahua 5231-ze cams, initially I only had 6 cams but when I eventually got round to installing the 2 remaining cams the switch would periodically flash the power warning LED even though on paper it was below the overall power output.

This never caused any issues with the 8 cams but I ended up replacing it with a HP 2530 POE+ switch as I had visions of it dying due to it running at its limit.