I'm curious as to possible alternatives?
There are none.
singletons are an anti-pattern
That implies that connecting to a database through a JDBC driver is anti-pattern.
Without being concerned about technical details a singleton is an object with a single instance. I've referred to singleton being an object since singleton is at runtime. Although there are different technics to implement singleton classes (e.g. private constructor, synchronised static factory method or ordinary static factory method with an inline initialised static variable and so on) an ordinary class with a single instance at runtime is a singleton.
Calling just once the constructor of a class through the entire code base of an application makes a singleton; would that be anti-pattern? No, it wouldn't.