Entering the world of "Lord Randall" can be a little disorienting. Being an older poem, with different literary conventions, it isn't necessarily that easy for us to access, if we read it like we'd read a more contemporary poem. It's not brimming with metaphors and similes, the way we've come to expect. Instead, it uses rather unexpected formal characteristics and bits of wordplay to make its point. So, what is that point? Well, at its heart, this is a poem about death, and the little figurative language that we find here has to do with the latter. And you can take everything else on the literal level.
Lord Randall's refrain occurs at the end of each stanza, and every time it does, its true meaning becomes clearer. While in the beginning, it seems that he actually means that he wants to go to bed because he's tuckered out from hunting, the answers that he gives to his mother's formulaic questions (an example of parallelism, by the way), all reveal that there's more going on than meets the eye.
The key to this revelation is the phrase "make my bed soon." It was something of a Medieval cliché to use this phrase as a euphemism for death. The "bed" referred to here can certainly refer to a normal bed, but it can also be a euphemism for a deathbed, or even a grave. In other ballads from the time, you'll often see speakers referring to beds, which usually means that someone's going to die. A particular cue is when a bed is "long and narrow" (like a coffin or a grave). A famous example of this trope occurs in the well-known folk song, "Barbara Allen." In our case, Lord Randall's weariness is not just due to hunting, but life itself. He's weary and "sick at heart" from his loved one's betrayal (oh yeah, and from poison), and he "fain wald lie down" not to sleep, but to die.