G’day Terry,
To understand rhyme a little more:  Ending each line with a strong beat (stressed syllable) as you have done here, gives the poem a general ‘strength.’ It’s kind of ‘bam, bam, bam.’ So it’s important to be mindful of the poem’s message. eg. one would likely not go ‘bam, bam, bam,’ in a love poem. (Well…maybe some real blokey types might… 

 ) This is called ‘strong’ or ‘single’ rhyme.
Where the rhyming lines of poetry end on a weak beat (unstressed syllable), it is called ‘weak rhyme’ or ‘double rhyme.’ It is a softer approach, & is therefore called ‘feminine rhyme.’
So it’s fine to go all strong, or all weak, or mix the two 
methodically.
Now this is where I think you are getting understandably confused: In weak rhyme, the last unstressed syllables (weak beats) of the words are the same, but (in the words used) the stressed syllables 
directly before the weak beats are strong rhymes. eg. ‘be-ing, see-ing’. ‘Be’ rhymes with ‘see’ and both end with ‘ing’. I know you are asking about the reverse, but the same thing applies....just in reverse. (Oh it's difficult to explain at times.) 
You have used ‘a-way’ and ‘way.’ As Neville noted this is ok….because…‘the’ (the way) aligns with ‘a’ in ‘a-way’. ie. ‘a-way’…. ‘the way’. Bob’s suggestion that you speak the lines out loud (or better still record them, & play back) will show you how ‘it sounds.’  But are you happy with something that is essentially ‘ok’? I think this discussion shows that you are not, & some adaptions so far are fine.  IMO it is simply a matter of having used two ‘too alike’ words that is weakening the poem. One way to correctly scan your work is to write it out as prose is written. One long line is best but hard to achieve on a pc. This is written in Iambic (one weak, one strong, one weak etc.) See how it scans. (A-1 actually)
I 
pause a-
while at 
Murph-y’s 
well and 
think of 
stor-ies 
it could 
tell of 
men from 
man-y 
walks of 
life, who’d 
stopped to 
wash the 
grime a-
way. The 
shear-ers 
and the 
min-ing 
types that 
rest-ed 
here and 
smoked their 
pipes, while 
swapp-ing 
yarns and 
bits of 
news, that 
they had 
heard a-
long the 
way.
See how you can identify a continual correct use of the structure here? Always one weak, one strong, one weak, etc. rather than becoming confused by individual lines of poetry?
The 2nd half of the last line is IMO becoming a tad ‘throw away’. It has much better opportunities. eg.
while swapping yarns and bits of news. What peace to muse upon that day. (Just a thought. I like Val's suggestion too.)
And also, to Bill. I think that Terry has achieved a 
very difficult thing here. He has 
successfully changed tense, & that is not easy to do. But Terry, take Bill’s advice in being mindful throughout the rest of the poem, that you keep doing it
 successfully or it could crash & burn. If you decide to wind up by returning to your initial 'place' (of standing, musing,) be sure to revert to the present tense.
Marty,
In your example, the only thing I can see that’s causing difficulty, is what does playing sport have to do with bracing a bridge? The structure is fine. (Iambic. one weak, one strong, one weak etc.) As above…‘of’ (of sport) aligns with ‘sup’ (sup-port). ie. 'of sport' sup-port'. Not knowing the rest of the poem puts me at a disadvantage. Perhaps it leads into why they’d then play sport?
‘They
 braced the 
Old Kul-
ar-a 
bridge with 
scaff-old  
and sup-
port what 
else to 
do in 
times of 
drought - but 
play a 
game of 
sport. 
Hope this helps.
Cheeers
Glenny