Agree with Maneki above in that it'd be very difficult to execute in 3 weeks strictly overland. BUT, if you're willing to take a flight between Luang Prabang and Hanoi (Lao Airlines or Vietnam Airlines), then you could at least get a 'good taste' of northern Laos and northern Vietnam.
You could cross from northern Thailand into Laos at the Chiang Khong to Huay Xai border and either take the 2 day slow boat down to Luang Prabang or overland via the trekking areas, taking the 4 hour bus from Huay Xai to Luang Namtha (perhaps an overnight trek), then you're a long day's bus ride to Luang Prabang. I recommend setting aside at least 2-3 days for Luang Prabang, including the time to visit Kuangsi Waterfall.
Then, after flying to Hanoi, one itinerary might be, bookend 2-3 days for Hanoi,
2days/1 overnight on a boat in Halong Bay,
return to Hanoi in time to take the overnight sleeper train to Lao Cai where you could visit Sapa and Bac Ha (or other interesting villages) 3N/2Ds.
I guess you could 'open-jaw' your SEA flights such that you fly into Bangkok and return to New Zealand from Hanoi but don't know about flights from Hanoi to New Zealand. If there are none, then you could return to Bangkok's Suvarnabhumi via Air Asia flight from Hanoi to BKK and then return home from there.
Odds and ends
weatherwise, March is a pretty warm to hot month with a build-up of haze due to swidden agriculture's burning of fields but very little rain.
Other option instead of heading up to Chiang Mai and across to Laos is to take the overnight train from Bangkok's Hualamphong Station to Nong Khai (unless the layover at Nong khai's station is stream-lined such that it's worthwhile to take the train all the way across into Laos),
visit Vientiane, Viang Vieng, minivan up to Luang Prabang and fly to Hanoi.
Several VIP bus companies go all the way into Vientiane from BKK (includes waiting time for immigration formalities at both borders) but the 'sleep' is not as comfortable as in the train's upper/lower compartments.
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