Yeah, you really can't get rid of political parties completely, or at least not without shredding the right to assembly and freedom of association. Although you could decrease or increase how much of a direct influence party leadership has on actual policy.
Personally I think the problem isn't even with political parties themselves, but with how our elections are structured. I personally would like to see the House of Representatives converted into a proportional representative system where people vote for parties rather then individuals. That way if half the vote in a state goes to one party and half the vote goes to the other party, then the State's delegation is split down the middle. Voters would know which individuals will be given a party's first seat, second seat, third seat...etc., and party leadership would be forbidden from changing the list of candidates within a few months of the election.
Then with single-seat elections (Senators, President, etc.), it would use a preferential system. So instead of voting for an individual, people would vote for a first choice, a second choice, a third choice, and maybe even a 4th or 5th choice. This would eliminate the spoiler effect, where a vote for a 3rd party implicitly supports the Democrat or Republican candidate someone despises the most, and allow people to vote purely on their own conscious for their first choice and vote strategically with their second or third. If 3rd party candidates start routinely winning 15-20% of the vote in the first tally, they might be taken more seriously in subsequent elections.