Polls taken to find the public's view on gun control overwhelmingly called for much tighter gun control, such as they have in every other industrialized country in the world. The house and senate overwhelmingly voted down the proposed measures (which had public support). This happened because the politicians were indebted to the NRA, and voted on the issue of gun control as the NRA lobby wanted them to. Their votes were purchased.
No one organization can have too much sway over the Senate unless they are acquiring their sway through illegal means, such as bribery or blackmail. If the NRA currently holds a strong sway over the Senate it is because they have earned the sway themselves. If anti-gun law groups hold the majority of the sway in the future, the same would go for them.
The majority of the senate is currently democrats, many of whom do not agree with the NRA's stance on gun control. I don't think that the NRA really influenced the vote on the recent bill. However, the NRA does make donations to both political parties in order to get support, which is not right.
Last I checked, the majority of the senate doesn't agree with the NRA, so thereby, no they don't. If they did have "too much sway" you would start to see some court rulings and other things being overturned, like the NFA and other laws that are outdated and we don't really need. But right now, the NRA just has enough power to block any new gun laws, which is good because we honestly don't need anymore.