Ron has some good points. You mentioned the key 'item' here: overflowing gutters. If your gutters are overflowing like you say, Ron's tips are certainly worth looking into. If the downspout is in sections, you might be able to disconnect it to see if it, OR the drain into which it connects, is clogged. (My point is that you want to check the part of the 'system' that takes the water to the street too (or wherever) for clogs.
If you decide to add additional downspouts, you may have to re-hang part of the gutters so it slopes 'the other way' as well as its current direction. If you need more help, post up. My best to ya and hope this helps.