Thad Cochran promises to deliver, but is Mississippi interested?

After six terms in the Senate, Sen. Thad Cochran, R-Miss., is making the case that he has the connections and clout in in Congress to deliver for his constituents. In a state like Mississippi -- which sees more than $3 in federal funding for every dollar paid in income taxes, and where federal funding has been critical after crises like Hurricane Katrina -- that may seem like an effective campaign pitch.

But instead of celebrating the funding that Cochran, a senior member of the critical Senate Appropriations Committee, has brought into Mississippi, conservatives there have lambasted the senator as the "king of pork." The label, along with the public's current dissatisfaction with establishment figures like Cochran, has left Cochran scrambling to win a seventh term.

On June 3, Cochran managed to win just 48.9 percent of the vote in the Republican primary while his conservative challenger, state Sen. Chris McDaniel, won 49.6 percent with 2,000 more votes. Since no candidate won 50 percent of the vote, Cochran and McDaniel are headed to a run-off election on Tuesday.

Ahead of the race, Cochran and his supporters are sticking with the message that his seniority in the Senate is a good thing. He's campaigning over the weekend with one of his most high-profile colleagues, Sen. John McCain, R-Ariz., while the Chamber of Commerce this week released an ad for Cochran featuring former NFL star and Mississippi native Brett Favre. "Thad Cochran always delivers, just like he did during Hurricane Katrina," Favre says in the ad.

McDaniel, meanwhile, has continued to campaign with anti-establishment, small-government allies. Last weekend, he campaigned with McCain's 2008 presidential primary challenger, libertarian former Rep. Ron Paul. A few days later, McCain's 2008 running mate, former Alaska Gov. Sarah Palin, reiterated her support for McDaniel.

"If the national GOP machine spent as much time fighting President Obama's disastrous and dangerous agenda as they've spent fighting that great conservative candidate Chris McDaniel, well, maybe they'd have been successful at stopping Obama's 'fundamental transformation' of our country," she wrote on Facebook to Mississippi voters. "Please send a message to the career politicians who sure seem satisfied with the trajectory our nation is on."

A recent poll from Democratic firm Chism Strategies shows Cochran and McDaniel are statistically tied, with Cochran leading 48 percent to 47 percent.

Marija Bekafigo, political science professor at the University of Southern Mississippi, said it's no surprise that the anti-spending message that conservatives have embraced nationwide is resonating in Mississippi, even though the state benefits so much from federal assistance.

"Mississippians are really just like other conservative Republicans," she told CBS News. "They are independent-minded thinkers who don't like taking handouts, don't like giving handouts, and if you're a McDaniel supporter, you think Thad Cochran is taking money and giving handouts."

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Furthermore, she said, McDaniel has effectively run on the message that the state has little to show for the federal dollars Cochran brings in. "We have all this spending that Cochran's brought us, yet we're still last in the nation in everything."

On top of that, Bekafigo said, some voters may see the 76-year-old Cochran "as having had his time, and people might be ready for fresh ideas."

Adam Brandon, executive vice president of the conservative advocacy group FreedomWorks, told CBS News that "Cochran should maybe take the opportunity to step down and let the next generation come in."

He said McDaniel would be part of a "new generation of leaders" within the GOP, along with Sens. Mike Lee of Utah and Rand Paul of Kentucky, who are putting special emphasis on shrinking the size of the government.

"If there's a split in the GOP between those who want to use government for cronyism or to keep K Street alive and flush, versus a Republican Party that stands on fiscal principle, that's a fight we've got to have," he said.

Whoever prevails in Tuesday's fight is likely going to have to win with fewer voters. A recent Fairvote.org study found that primary runoff elections typically see a significant decline in turnout. In fact, in 168 primary runoffs between 1994 and 2012, all but six saw a decline in turnout.

That could benefit McDaniel, who enjoys a more energized base. "He seems to have more momentum behind him, and his voters are obviously fed up with Washington establishment politics," Bekafigo said. At the same time, she noted that Cochran appears to have an edge among older voters, who are more dependable.

"While McDaniel may have the momentum, he has to get them to show up, or we're going to have another close election," Bekafigo said.

Third parties backing both Cochran and McDaniel in the past two weeks have continued to pour money into the race, and even some Democrats are getting involved to support Cochran. The senator has said he's happy to accept support from Mississippi's Democratic voters, who are free to vote in the runoff as long as they did not vote in the June 3 Democratic primary.

"We don't encourage anyone to miss out on voting. We think the more the merrier," Cochran said at a recent education roundtable.