Please continue doing your backups!
Backups are Disaster Recovery, yes…but not HA.
Some will argue with this (in the comments most likely), but I broadly define “High Availability” as a system that can recover in seconds or minutes at most. Sometimes that is automatic, sometimes manual.
I agree that backups are for DR, not HA. I’d consider log shipping an option for both HA and DR, albeit one that requires manual failover (or rigging up a script that performs the failover for you).
I disagree about replication as an HA solution. Yes, you do need to make sure that everything can replicate, but if your publisher goes down, the subscriber can continue and your data is still available for use. And if you’re a complete masochist, you can use merge replication to allow writes to continue while the publisher is down. Cleaning up after that is a mess, especially if you end up with a bunch of conflicts, but High Availability doesn’t mean Easy Mode.