As the 2016 presidential campaign ramps up, it may seem like deep-pocketed, often anonymous political financiers are growing more influential by the day. The most recent CBS News poll shows that voters -- Democrats, Republicans and independents alike -- are tired of it.

Leaders in Washington aren't doing much about it, but at the state level, lawmakers from both sides of the aisle are considering ways to give voters more information about who's bankrolling expensive attack ads and shadowy robo-calls. At least 125 transparency bills were up for debate this year in more than 30 states, according to research from the Sunlight Foundation. Most of them aren't going anywhere, but even in deep-red states like Texas, politicians say the issue is an urgent one.

"I believe that it's very important to make sure that people are able to align the message and messenger and make an informed decision that best serves the public," Texas Rep. Byron Cook, a Republican who chairs the State Affairs Committee, told CBS News. "In a state like Texas where we don't have meaningful restrictions, where people basically give whatever they want to campaigns, it becomes even more important that we address this kind of issue."

Specifically, Cook is talking about curbing the rise of "dark money." Thanks to a series of recent court rulings, certain nonprofits are playing a growing role in elections. These tax-exempt "social welfare" organizations -- in the 501(c)(4) category -- are allowed to spend unlimited sums of money on politics as long as politics isn't their main focus. Furthermore, in most cases those organizations aren't required to name their donors.

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"If we don't address this issue, I think it will just be a matter of time before high net-worth contributors don't give directly to [political action committees], they'll simply write their checks to 501(c)(4)s because they can do so anonymously," Cook said. "That really has a corrupting effect on the democratic process."

Cook led a bill through the Texas state House that would have required those "social welfare" groups disclose the names of donors who contributed at least $2,000 specifically for campaign expenditures. The bill passed with overwhelming bipartisan support in the state House, but it died in the state Senate just before the legislative session ended.

"I was hoping Texas would be proactive and address this before we have a big issue, which we will eventually have," Cook said.

Cook is convinced that dark money will lead to corruption after watching the dramatic case that unfolded in Utah.

In 2012, state official John Swallow was elected Utah Attorney General after allegedly exploiting a network of shadowy nonprofit organizations to attack his opponents. Nonprofits with vague names like "It's Now or Never" were allegedly created by the Swallow campaign for the purpose of keeping his backers in the payday lending industry anonymous. Swallow quickly came under scrutiny and resigned after less than a year in office. He was arrested last year and will be arraigned next month on 14 charges, including accepting or soliciting bribes and failing to disclose conflicts of interest.

Before Swallow was arrested, the Utah legislature formed a committee to investigate his alleged misconduct. It concluded that "Mr. Swallow hung a veritable 'for sale' sign on the office door that invited moneyed interests to seek special treatment and favors." Swallow solicited contributions, the committee wrote, that "funded nearly untraceable negative attacks."

State Rep. Jim Dunnigan, the Republican Majority Leader in the Utah House, told CBS News that the investigation led to a series of reforms in Utah.

"We made several changes that I think really improve things, and now I think we should let those take effect and see if any additional changes need to be done," Dunnigan said.

The state ratcheted up reporting requirements for 501(c)(4)'s and tightened conflict-of-interest reporting requirements for candidates, among other things.

Some public interest advocates in the state want to see further campaign finance reforms, but Dunnigan said that the transparency rules passed are a good start.

"We layered on more campaign reform this session, and I do have a little concern we get to the point where it's going to be hard for the good actors to stay in compliance," he said. "I support the ones that shine the light of day on what's going on, but don't get so complicated you need to hire someone to follow you around and make sure you stay within the lines."

Montana is another typically-conservative state that's successfully passed transparency reforms. Earlier this year, Democratic governor Steve Bullock signed into law a "dark money" bill that was shepherded through the state legislature by two Republicans.

Republican Rep. Frank Garner, who guided the bill through the state House, said he started paying attention to dark money after the group Americans for Prosperity-Montana started attacking him.

"You really don't know you need a raincoat until it's raining," Garner reportedly said at the bill signing. "That's kind of what happened to me this session. As a freshman, this was not something that was on my radar screen."

Democratic states like California and New York are also implementing dark money disclosure rules.

At the federal level, transparency in elections has been a Democratic cause. Since the Disclose Act has gone nowhere in the GOP-led Congress, activists are putting pressure on President Obama to unilaterally require companies with federal government contracts to disclose their political spending.

However, Texas Rep. Cook insists the issue isn't really a partisan one.

"Abuse of this will come from all sides," he warned. "It won't be exclusive to any side, and the public has a right to make an informed decision" during the election season.