Republican gubernatorial candidates are hitting up donors, crisscrossing the state and popping up in surprising places as they claw for any edge in a crowded and wide-open GOP field.

The candidates face an unusually testy, expensive campaign that features a millionaire and a former House speaker, and which will not be decided at a state endorsing convention in late May. Several candidates have said they will not honor their party’s endorsement and will press on to the Aug. 12 primary.

Their first test behind them — a caucus straw poll — the race now begins in earnest for Republicans looking to unseat DFL Gov. Mark Dayton.

“It’s ’round-the-clock,” said Jeff Johnson, who must balance his campaign with his duties as a Hennepin County Board commissioner. Johnson is on a particularly limited clock, as one of those who has pledged to drop out if he fails to get endorsed.

In a bid to win potential delegates, state Sen. Dave Thompson is already promoting his running mate; former House Speaker Kurt Zellers served as a keynote speaker for a wholesale beer sellers event; and Orono businessman Scott Honour has a New York City fundraiser set for next month. Along with Johnson and former House Minority Leader Marty Seifert, the five are locked in a grueling schedule of drumming up cash, making late-night calls to activists and popping up at diners and VFW halls around the state.

“It is not only about delegates, it is not only about fundraising, it’s not only about media,” said Seifert, in the midst of a busy day of campaigning. “It’s about being able to pull the hat trick of balancing your time with all of those different groups.”

Minnesota’s GOP field faces a daunting challenge: Convince activists they are hard shell conservatives while still persuading big-money donors and average Minnesotans that they have the right message, smarts and raw hunger to beat Dayton, who is already building a formidable re-election campaign.

Dayton heads into the campaign season buoyed by a strong economy, low unemployment and a state budget surplus approaching $1 billion. He already has raised more than $1 million, hired four full-time staffers and last week rolled out his new running mate, his chief of staff Tina Smith.

But Republicans see a large enough opening to defeat Dayton, an heir of a family department store fortune.

They hope that lingering outrage over the bungled rollout of MNsure, the state’s health insurance exchange, and a strong national tide favoring Republicans will allow them to sweep past Dayton, even with the strong political apparatus behind him.

“He’s beatable,” Zellers said. “One thing Republicans haven’t done very well is have a vision that brings people together. We’ve got that.”

Several Republican strategists say Dayton is a savvy and tough-minded candidate not to be underestimated. Candidates say they must win the primary and then quickly prove they have a winning strategy.

“If people have a sense this governor is beatable, there will be the resources put into the race,” said Thompson, who is focusing on fundraising and reaching out to GOP convention delegates.

But, he said, if a poll comes out in August showing the GOP candidate 20 points down, “the money won’t come in.”

With a wide open and diverse field, each candidate is working to overcome or smooth over possible weaknesses.

Johnson lacks statewide name recognition and so far has not proven to be a gifted fundraiser. Zellers’ reign as House speaker abruptly ended when voters tossed Republicans from leadership, which still rankles many activists. Thompson lacks name recognition and is light on funds. Four years ago, Seifert lost the GOP nomination to Tom Emmer, who in turn lost to Dayton in what became Republican’s best election cycle in years. Honour has no political experience and has already burned through more than $500,000 in campaign money, which operatives say is a novice mistake.

Johnson is telling Republicans that he is conservative enough to win the GOP endorsement, but has a broad enough appeal to win a crucial slice of independent and Democratic voters.

He rises at 6 a.m. and doesn’t go to sleep until nearly 1 a.m., his day a blur of commissioner duties, endless calls and e-mails sent to activists and potential contributors.

“For us to win, we have to draw a contrast with Mark Dayton,” Johnson said. “But that can’t be the only thing we do. We have to share a positive vision that is relevant to everyday Minnesotans who aren’t Republicans.”

Zellers, a Maple Grove public relations consultant, puts upward of 5,000 miles a month on his car for the campaign. He has a three-ring binder full of names of donors. He spends nearly every mile on the road talking to activists, reporters and potential contributors. In between, he coaches football, baseball and hockey teams where, he confides, he gets his best campaign anecdotes talking to parents.

Thompson is working relentlessly to win over likely GOP delegates. Like Johnson, Thompson’s gubernatorial dreams will be snuffed out at the state convention unless he wins endorsement. Thompson has said he wants the full weight of the party behind him in the certain primary ahead.

Seifert is trying to present himself as a more moderate version of the guy who ran against Emmer four years ago. Seifert had left the intensely political role of House minority leader to focus on the governor’s race. With a strong Tea Party flavor in the last GOP race, Seifert painted himself as a fiscal and moral conservative, and a skeptic of any human role in global climate change.

“Working in the Capitol with partisan people all the time, there is a different attitude,” said Seifert, who was executive director of the Avera Marshall Foundation before leaving for his latest run for governor. “Four years in the private sector, working side-by-side with nonpolitical people, gives you a different perspective.”

Honour, who can tap a personal fortune for his campaign, is spending a lot of time just getting his name out — sometimes in surprising ways. On Friday, the group Singles for Honour co-sponsored a singles event at Mall of America to benefit St. Joseph’s Home for Children. Promoted as a life-changing, matchmaking event, the evening featured a relationship expert and celebrity matchmaker.

A successful effort will take money, too, which is why Honour is holding a fundraiser in New York.

“Getting the fuel in the tanks is critical,” Honour said Friday. “We are gearing up toward having the approach laid out, and having the right team and the resources to win.”