Those may sound horrible, but they are nonetheless relatively common when talking about playground swings.
I went swinging on the swing yesterday.
Such redundancies are sometimes inevitiable when the same word functions as both a noun and verb. Here are some other examples that sound just fine:
I had too many drinks when we went out drinking last night.
I went fishing but caught no fish.
That said, these may be inevitable, but they aren't necessarily unavoidable. Such sentences can usually be rewritten if the redundancies bother you:
I had too many drinks last night.
I went fishing but caught nothing.
In the case of the swing:
I was playing on the swing yesterday.
I was using the swingset yesterday.
Sure, you could say:
I went swinging yesterday.
but in the case of swinging, you might want to clarify that (see Collins #3).