If your Russian accent is strong, reducing it is an important thing to do, imho.
It does not sound particularly pleasant, and many ppl (especially older folks who lived through a Cold War, and are now in a position to give you job interviews and stuff) tend to be suspicious towards such speakers (unlike certain other accents, like British/Australian for example which has an inverse effect). Of course this is not universal, but it is still common enough.
On the other hand, a _slight_ accent is usually OK and some people will find it cute.
In addition to songs, I'd mention audiobooks - some of them are read very professionally, and it makes a _huge_ difference when you are trying to imitate or learn new sounds. George Guidall (e.g. American Gods by Neil Gaiman here, a great book btw) was particularly good in my opinion.
And yes, a "language coach" may help a lot - but only if she is good at it and knows what she is doing, and speaks very clearly.
A random native speaker is not going to help too much. Although if your accent is really strong she may still help, at least by locating things that you need to work on. But at some point people without training will not be able to do this anymore. They'd just be able to tell you that something is not right, but not exactly what, because some issues are rather obscure.