Do I need a new backup disk or should my current setup keep going deleting old
stuff as it becomes full?
It worked in the olden days.. when you had OS like Snow Leopard that was most current Mac OS for about 4 or 5 years..
Now an OS lasts 12months.. if that.. and after 5 or 6 patches most 1GB or more download which is expanded out to 4 or 5GB Time Machine has no idea which files can be deleted and which cannot.. hence your Time Machine backup fills the disk and stops. It has been like that for ages now.. but Catalina made it hugely worse.. since it likely started a whole new backup.
If I need a new one what is a good replacement for the Airport Time Capsule that includes the distribution function or do I need to do that in a different way.
First of all a TC that had a 1TB disk is at least 9 years old.. and could be up to 12 years old.. so it is well and truly time to replace the TC regardless.
No manufacturer makes an exact equivalent to the Time Capsule.. but yes you can buy a router with 5 ethernet ports and will work with Time Machine via a USB hard disk. Synology RT2600AC is the best of them. They are equally fast as best TC disk was, and reliable enough for backups. (Nothing for Catalina is 100%)
A better alternative is Synology NAS. Mostly you won't have multiple ethernet ports unless you spend a lot of money.. even cheapest NAS is still expensive cf a router and USB drive. What the NAS is missing in ethernet you can make up for with a $20 8 port gigabit switch.. so no major problem there.
The big advantage of NAS is multiple disks, automatic backup of the NAS itself, huge selection of excellent apps including things like iTunes. AND FAST.. it will saturate gigabit and even more than that if you go up the scale. That is about 3x faster than TC or Router with USB drive. It also will last longer so good investment. (When I say TC I am NOT referring to yours.. which is already at ancient technology status).
Now one other point.. for backup to an iMac. a USB drive plugged straight into the Mac is still the cheapest fastest and most reliable method especially with Time Machine in Catalina.
Other routers do support Time Machine.. but not so reliable. I recommend the Asus routers.. I use an 8 lan port industrial type RT-AC88U .. but I use it with Carbon Copy Cloner.. which works better than Time Machine.. how much better I will tell you in due course.. I am still testing.