One thing you might try is writing a good bass line to your melody. Since the melody moves primarily by steps, it will be important for the bass to move in contrary motion in most instances in which the melody moves by step (2nd). Rather than harmonizing in the left hand with root position chords, try first harmonizing in "keyboard style," i.e. bass voice in left hand, all other voices in right hand. When melody and bass together suggest a first inversion chord expand the right hand from closed position chords to an open "1-5-8" voicing. This technique, combined with "keyboard style" and contrary motion when bass is moving by step, will help you to avoid parallel fifths and octaves. Avoiding these consecutive intervals is key to maintaining a consistent texture of a given number of voices.

Points of clarification, EAF- you're wanting a homophonic setting?
Are you "married" to the chord assignments given originally?

My humble recommendations:

(1) Ditch the chord progressions in your attempts to write singable 4-part harmony. This is a technique better (and even then, ill) suited to strummed accompaniment in song writing. Indeed, in the example given, one gets the impression from the assignment of half-note chords on beats one and three, that the melody notes of beats two and four are unimportant and mostly relegated to passing note status. This is not good practice for solo song writing, and it is even further from being a hallmark of good 4-part hymn writing.

(2) Starting with a bass line (moving generally with the same tactus as the melody) is sometimes (but not always) a good start, in that it forces the hymn composer to think about 2-part counterpoint. However, in the course of composing a richer 3- or 4-part structure, almost invariably either the original bass line will likely prove inadequate in a few - or several - places, forcing you to change it to something else or perhaps even altering the melody (if that is permissible).

(3) The previous point points to the necessity of understanding that, in good hymn composition, one must face squarely the issues of 4-part harmony and counterpoint, striving for a balance amongst voice leading in individual parts, vertical integrity of the harmony, and the overall harmonic rhythm and melodic flow of all four parts. This is not always to achieve in practice, and many attempts will be assigned to the wastebasket or (do not) recycle bin.

(4) If you are somewhat inexperienced with 4-part writing and have progressed to the point of being able to construct a singable 2-part texture (not necessarily melody-bass, since 2-part counterpoint is less concerned with harmonization), then you might do well to proceed to 3-part writing instead of jumping directly to 4-part writing. For one thing, SABar compositions often are better suited, for better or worse, to the singing abilities of present day choirs and congregations.

(5) Whether writing 3-part or 4-part hymn music (or 4-part music augmented with one or two descant parts, as I am wont to do at times) you have to be prepared – again and again, always – to modify your previous work with a view to improving the result or even to fashioning entirely new harmonizations (that may, in the end, prove to be better - or worse - than the original).

(6) Seek was to discipline and ennoble yourself in hymn writing. Study the venerable and classic hymn tunes, as well as and how they have been harmonized. Working with good hymn tunes, do try to compose descants for them, altering the lower voices if necessary to achieve acceptable or even good results. And don't shy away from composing a new harmonization of a standard hymn tune – it can be a hardy task that deepens your own abilities. Compose 3- and 4-part music that is not necessarily destined for or even intended for hymn or choral music. Start with shorter pieces or meters ... a Short Meter (66. 86) tune is generally easier to work with than a Long Meter Double (88. 88. 88. 88) tune ... in the same way that a couplet or even a limerick is far easier to pen than a sonnet.

Finally, if it truly is in your blood, don't let the discouraging moments and inevitable lapses or failures get the best of you. Create what you enjoy – and enjoy what you create.

The penchant to write hymns or verse
A blessing can be, or a curse.
  When showing my tune
  To someone last June,
She said that I couldn't do worse.

Messiaen set seven of them!

Thank you all for the advice!

What do you mean? Keep in mind, please, I only know High School-level music theory and am at heart a trombone player. (My instrument doesn't need so much a hand as a disfigured claw.)

You might find it worthwhile to read the classic counterpoint book by Johann Fux, ideally with a coach to guide you through exercises.

Just for fun, here's one of my counterpoint exercises this week: here.