We know Dylan Thomas was very much inspired by Romantic poetry. "And death shall have no dominion" is no exception to that trend. It's perhaps a near perfect example of just how much he loved William Blake and Co.
The only difference with Thomas is that he's not so much a stickler for keeping with a particular form or meter. Instead, he likes to play around with meter and rhyme in a way that kind of flexes those modern poetry muscles. All the playing around makes "And death shall have no dominion" fall in the free verse camp while still paying homage to those Romantic juggernauts with relatively regular lines. Then of course we have those slant rhymes that make the poem sound super modern. Lines 1-3 highlight Thomas's rhyming style with all those N sounds in "dominion," "one," and "moon." They're not perfect rhymes like our Romantic poets would prefer, but they do rhyme those consonants together nonetheless.
A little later, we do have an instance or two of perfect couplets like in lines 11-12 ("sea/windily") and in lines 15-16 ("two/through"). So it's not like Thomas is totally against perfect rhymes. It's just that he prefers not to overdo it and instead throws a few in when they work with what he already has. He's not one to force a perfect rhyme unnecessarily.
The poem is also highly lyrical (song-like), which hearkens back to the Romantics, too. The refrain is kind of like the chorus in a song and all the parallelism we see in lines 6-8 makes his ideas hard to forget ("Though they…). To top it all off, the poem also looks highly organized with three stanzas containing nine lines each, with the refrain beginning and ending each stanza. So Thomas is pretty much demonstrating that he's got those form and meter chops, but he's a bit too modern to devote himself to one kind completely.