Not exactly. There are two things that win elections: Money and votes. And money only because it can be used to buy votes.
When the major parties agree on an issue it's because that issue has a stable equilibrium: Either the parties both side with the electorate or the monied interests because the number of vote-equivalent dollars gained in changing positions is not greater than the number actual votes lost by doing so (or vice versa). So you don't see many politicians seriously advocating legalizing murder or drastically slashing the defense budget, respectively.
But all you have to do to get them to change is to make the number of votes they lose by siding with corporations on a particular issue greater than the amount of money they get by doing so. You don't even have to get the parties to split -- most of the issues they're split on are intentionally wedge issues or detail-free hogwash like "small government" or the vague notion that rich people should pay more taxes. You can actually get them both to start siding with the interests of the people on a particular issue, you just have to make more noise than what the opposing interests are paying in dollars. It has happened before. The establishment of the EPA, the breakup of AT&T, the defeat of SOPA, it's a pretty long list.
But it only happens if we make it happen. We have to make them understand that if they don't do what we want, they lose more votes than the money they're being paid can buy them back. Getting 100,000+ voters to sign a petition that brings the issue to the attention of lawmakers may not be sufficient to achieve that on its own, but it ain't a bad start.
Out of 435 house representatives, each election every two years, only a dozen or two seats will change party. At best. That often throws the balance in congress towards one party or the other, but the vast majority of congressmen can be corrupt as fuck and never worry about losing their seat because they are in "red" or "blue" districts.
If you don't live in one of the few turbulent districts with political discourse, nothing will change and the congressman has no care for his / her constituents because they can't be displaced, and the constituents are brainwashed into an us vs them mentality and local culture of blaming everything on the "other guy" even when their rep is actively voting against them.
In addition to that, gerrymandering just pushes the real campaign into the primaries for those seats. If your representative's party has a hard lock on your district's seat, do what the Tea Party has been doing and primary the incumbent. And fewer people vote in the primaries, so each vote you can shift from one candidate to another counts for more there.
But you can get the result you want through the primaries. Run your candidate against the incumbent in the primary and keep emphasizing your specific issues and publicizing the incumbent's corrupt stance on them -- even if you don't always win, you may be able to get the incumbent to change positions.
This is known as Duverger's Law:
http://en.wikipedia.org/wiki/Duverger%27s_law
There are a number of meaningful counterexamples, perhaps most notably in the UK, where Lib Dems have received a significant number of seats since 1974.
These discussions tend to obscure the degree of differences within a given party in the US. (ie, Zell Miller and Dennis Kucinich once shared a party. Though Kucinich was given an "F" by the NRA on gun issues, Zell was later elected to the NRA's board of governors. Kucinich was also a strict vegan, while Zell once threatened to shoot Chris Matthews. You could run similar comparisons by bringing in other Dems, you could do the same on the other side of the aisle. ie, not many Republicans would consider marrying one of the Kennedys or praise Obama's post-hurricane support, but a couple would.)
Any two-party system with a diverse electorate will eventually have to allow substantial variation in candidates and viewpoints, or it will be inevitably reduced to regionalism and obscurity.
A many-party system will eventually force smaller parties into large coalitions to attempt any governance at all.
In short, a two party system and a many-party system are not substantially different in the number of political viewpoints they entertain, nor in the types of compromises required between those factions to result in political action.
If you're not already aware, I would encourage you to check out Lawrence Lessig and the Rootstrikers organization which is aiming to reduce the influence that money currently has on elections.
Question: Can you recommend any particular organizations that are employing your above strategy of targeting primaries of contested districts to defeat soulless "bought and paid" incumbents?
There isn't a choice. It is why I vote libertarian. But the root problems are deeper than that, mainly that you have individual citizens voting for too many people they have no empathy or commonality with, and that politics is a profession rather than civil service.