Fulton was on 41 while Williamson was 16 not out as the hosts were still 134 runs from making England bat again and needing the rain predicted for late on Sunday to arrive earlier than expected to save the match.
While showers are predicted late on the fourth day, New Zealand's MetService has forecast persistent rain for the Wellington region throughout Monday.
England captain Alastair Cook had enforced the follow-on after bowling New Zealand out for 254 shortly before tea, with the hosts 12 runs short of the required target to put the tourists back in to bat and well adrift of England's 465.
Stuart Broad had rattled through New Zealand's lower order with figures of six for 51 and Bruce Martin was left stranded on 21 not out before Cook enforced the follow on at tea.
"It's not often enforced to be honest because the bowlers tend to like a bit of a rest and it's good to get their batsmen back out in the field and build a big lead with scoreboard pressure," Broad told reporters of the reasoning behind enforcing the follow on.
"With the weather around it's unsure how much cricket will be left in the next two days, that was the only reason behind it."
The hosts began their second innings under intense pressure from Broad and James Anderson but left-arm spinner Monty Panesar got the only breakthrough when Hamish Rutherford was brilliantly caught by a diving Ian Bell at leg slip for 15.
Fulton and Williamson, however, survived through to stumps to ensure the game went into a fourth day and gave local fans a slim hope of the team salvaging a weather-influenced draw, though New Zealand wicketkeeper BJ Watling said his side still felt they could pull out an improbable victory.
"Their bowlers have bowled quite a lot of overs now so the longer we keep them out there hopefully they get tired and start loosening up and giving us balls to put away and then we can put pressure back on them," Watling said.
"The wicket's a good track. I think we are obviously disappointed with our first innings effort. We needed 350 at least on that so if we can rectify ourselves in the second innings and put them under a bit of pressure (by scoring) 350-400, a lead of 200 on this track could be defendable."
New Zealand had looked like they could avoid the follow on at lunch with Brendon McCullum in an obstreperous mood, having reached his 27th test half century shortly before the break.
The New Zealand captain quickly advanced to 69 before Steven Finn got some extra bounce, caught the outside edge of the bat and Jonathan Trott took a simple catch at second slip.
McCullum's dismissal, immediately after he and Watling had brought up their 100-run partnership, left New Zealand on 189 for six.
A reckless shot from Tim Southee, who inexplicably fell into the trap set for him by hooking a short Finn delivery directly to Broad at fine leg for three, left the hosts on 197-7 and teetering on the brink of total collapse.
Watling upped his scoring rate after Southee's dismissal and brought up his fourth test half century with a punch in front of point off Finn for two runs.
He was supported by Martin in a 42-run partnership before the second new ball ended their resistance when Watling (60) feathered a catch to Matt Prior off Broad.
Neil Wagner became Broad's fifth victim when he fell in exactly the same manner for a 13-ball duck, then the tall fast bowler snared Trent Boult for two to finish with his third-best career figures.