All you are describing is a regular NAT router.
If you have a router with a DSL modem you deserve a medal if you can find one that won't do NAT.
Maybe one that doesn't do NAT might be a PCI DSL modem or USB modem I once saw, and if they don't do NAT, then even then perhaps some will do NAT.
Looking for a DSL router that does NAT is like looking for a dog, but not just any dog, a dog with a tail. It's almost a joke question. An you know jolly well that you're looking for a router 'cos you've mentioned the word "router" in your title.
And when you are that new to networking, you can go into any big computer store and ask and get info.
And you're talking about subnetwork and "address space" and using the terms correctly. Like a great student of networking that is not familiar with what regular routers do. If you ever set up your own router and maybe one for your cousin, and have seen your friend's, then you'd know what you have how normal what you have is.
The hardware you want, is as you wrote in your title. A Router. You probably want one with a DSL modem in it..
You even managed to figure out that your router has a network switch inside it. That is absolutely correct.
It is a router but one might "more accurately" call it an "network device" as it does so many things and its routing isn't that developed. It also does "switching". It may have a modem in it if it can connect directly into the wall. If it is wireless then it has a wireless access point in it as well. It has a DHCP server which is handing out IP addresses in a particular address range. It runs a DNS server as well. An HTTP server for you to view its web interface. Its router aspect is not very impressive, its router aspect has only 2 ports or 2 sides, the LAN side and the WAN side. The LAN side has a network switch attached. Essentially the router only two subnets. One for WAN and one for LAN. The majority of home routers and small business routers only support one subnet for LAN and one subnet for WAN. For a more advanced router one might have to one made by cisco or juniper. So it'd take some effort for you to find a router that doesn't do what the one you have does.
There are some rare NAT routers that have only one socket for the LAN. But hardly anybody would buy them. They're hard to find. And i've seen cable modems that don't do NAT though i'm sure many do NAT. But DSL routers, they'll do NAT unless you've really made an epic hunt for one that doesn't, in which case it'd be interesting if you share the name of the one that doesn't(and even then you may find that it does).
Sometimes routers have a "bridge" mode. Bridge is a switch functionality. Meant to turn the router off, probably does, and that mode turns NAT off too. Some may have an option to turn NAT off. But they'll have NAT on by default.