Does WiFi still work if I use my router as a switch?

Yes. By turning off dhcp and wan but leaving the WIFI on you are in effect turning it into an access point.

You will likely want to move its management IP to something other then the default, and make sure its in the same subnet as the new routers LAN range, and as you stated ensure DHCP is off.

If your new router includes WIFI, if you have same SSID and password but different channels you will also have almost seamless roaming between the two.

I have just done this in my house. I have a new router with a few devices connected (both wired and wireless), and I have my old router wired to it and set up as an access point, with additional devices connected to it (both wired and wireless).

Besides changing the settings on the old router to work as an access point, I also had to connect the cable coming from the new router to one of the four "outputs" on the old router, not to the "input" labelled "WAN". This seems counter-intuitive, but that's how things work if you use a router as an access point.

I also gave the access point a fixed IP address in the settings of the new router; I'm not sure whether this is strictly necessary, but then you can easily reach the access point's web interface to change its settings.

Many routers have a separate Access Point mode. Setting the router into Access Point mode allows it to be both an AP and a switch, I have checked and verified this. I don't think the switch behaviour can be disabled at all in these routers.

I can also expand a bit on @guest's point about moving the router-facing cable from the WAN port to the LAN port. Most such wifi routers effectively have a

1. Routing block, connected to the WAN port on side and to one port of the internal switch on another.
2. The internal switch itself which is very nearly unmanaged(unmanageable?) - it has virtually no parameters to control.
3. A WiFi access point "bridged" to the internal switch via another one of the latter's ports.
4. The management interface which may sit within (1) or be separate.

Any packet arriving at the internal switch is simply switched blindly. If the destination turns out to be accessible via the AP port, the AP converts the packet to 802.11 from 802.3 format and broadcasts it (and vice versa). Much of the configuration controls either (1) or (3), individually.

The reason I know this is that internet has mostly worked for me on a few occasions even when I or someone else had unknowingly plugged in the WAN (ethernet) cable into one of the LAN ports instead.

@davidgo's suggestion of keeping it in router mode but disabling WAN and DHCP is not the same as AP mode since several Layer 3 functionalities like NAT(which you may not be able to disable individually at all)/port forwarding/UPnP and firewall would still be running on the router consuming CPU cycles and RAM. So if you have an AP mode, use it.