Possible to connect each switch to the router and to each other? Yes, but it isn't as simple as just connecting them.
To get the operation you describe ("internet traffic will go directly from either switch to the router, but LAN traffic can go directly in-between switches")? No, not without complicating the configuration of end devices and making maintaining the network exponentially more difficult.
The interfaces on the router will either be L3 interfaces (which would require separate IP addresses per interface) or L2 bridged interfaces (which is not recommended - see below).
With L3 interfaces, each interface would have a different IP address. The devices connected to the network will only have one of them configured as a default gateway and all internet traffic would flow through that one L3 interface. The only way to get it to work as you describe is to do something crazy like manually configure GW1 on devices connected to SW1 and GW2 to devices connected to SW2.
With L2 interfaces, spanning-tree should work to provide a loop free environment (L2 loops are bad). This means that all the traffic will either flow from switch to switch then out one router interface or all traffic between switches will flow through the router.
Personally, I would connect both switches to the router and to each other, but simply for redundancy. Use L3 interfaces and run VRRP between the interfaces. Each L3 interface has it's own IP address and then "float" an IP between them, which would be the gateway. If you have a problem with one link to the router (one of the switches dies, cable gets disconnected, etc), then internet traffic will still be able to flow out the other L3 interface.
Finally for the note on bridging; bridging is discouraged by the vendor (at least for some Edgerouter platforms):
It is generally discouraged to enable bridging because traffic which is bridged is not hardware-offloaded, and will cause a decrease in performance. It is highly advised to use a dedicated switch connected to one of the LAN interfaces to allow for multiple ports of connectivity to the LAN rather than bridging.