What an interesting question! David gave a great answer, but i think i can add some context and flavour which may help further visualize.
Your ear is not leading you astray. As David noted, the edges that separate the two are not clean, and infact clever composers will intentionally blend the approaches.
The Core Distinction: Horizontal vs. Vertical
The simplest way to think about the difference is to consider the primary focus of the music.
* Homophony is primarily vertical. The main goal is to create a succession of chords. All the parts, including the main melody and the accompaniment, move together rhythmically to form these chords. The main interest is the harmony that happens at any given moment. Think of it as a series of pillars holding up a roof (the melody).
* Polyphony is primarily horizontal. The main goal is to create interwoven, independent melodic lines. Each voice has its own distinct melody and rhythm. The harmony that results is a byproduct of these independent lines crossing and interacting. Think of it as a conversation where different thoughts are being shared on the same topic, simultaneously.
Why Songs (hymns in this case) Feel Complicated
Musical textures aren't always one or the other. They exist on a spectrum. What you're describing is a perfect example of something called chorale-style homophony.
* Homophonic because all four voices (soprano, alto, tenor, bass) generally sing the same syllable at the same time and move with the same rhythm. This creates a clear, vertical chordal structure. The soprano carries the main melody.
* Polyphonic elements because good composers write the inner parts (alto and tenor) wifh their own melodic shape and contour, a concept known as voice leading. These inner lines are distinct but supportive in nature. You correctly identified their inherent weakness as a standalone melody.
When you're listening to a piece of music and trying to identify the texture, ask yourself these questions:
Am I listening to one primary melody that everything else is clearly supporting? That's a strong indicator of homophony.
Or is my ear being pulled between several different, competing melodic lines of equal importance? That points toward polyphony,
Do all the parts move together in a rhythmic lockstep? This is the most telling sign of homophony.
Or do the parts have their own distinct rhythms, starting and stopping at different times and creating a complex rhythmic tapestry? That's the essence of polyphony.
Can you hum an inner part ... say a bass line?
In a Bach fugue for example, the bass line is just as much a "melody" as the top line. In most homophonic music, the bass line exists purely to support the harmony and wouldn't be very interesting by itself.
Lyrical independence is not the defining feature of polyphony, but it is a common characteristic in vocal polyphony (like a Renaissance motet).
And to your final question: Yes, many pieces of music have characteristics of both. Composers frequently shift between textures for dramatic effect. A piece might have: A powerful, homophonic opening, then a complex, polyphonic development section followed by a return to a grand, homophonic texture for the finale.