If I might add my 0.02 to this discussion, I have heard great things about the AM and (especially) FM reception offered by the CCRadio SW (a.k.a. Kaito KA2100, a.k.a. Redsun RP2100). Some say it's the best portable out there for dxing! However, I've never been able to test that claim. The S350 is much more prone to overload from strong local signals and it's analog tuner is not as crisp as that of the CCRadio SW. As a dx machine, pretty much everything I have read points to the CCRadio SW as being the better of the two.
The radios that I can tell you are great dxing machines for AM and FM bands are the Eton E5/Grundig G5 and their cousin, the Kaito KA1103/Degen DE1103. All offer outstanding sensitivity and selectivity on the AM and FM bands. And, all are under $150. I'll agree that all are much better at picking up weak signals than the G4000A, though it's a much closer contest with SW bands.
With regard to the Eton E1, my take on it is that it is an awesome unit for SW dxing, but that it will not outperform any of the radios that I mentioned in dxing the AM/FM bands without an excellent external antenna. As it's less prone to overload than smaller units, the E1 is better able to take advantage of an external antenna than any portable would. But independent tests have indicated that the E1 is no more sensitive than the E5/G5 or the CCRadio SW - and that it might in fact be a little less sensitive. It is more selective and has a lower noise floor than the smaller units.
At 4 times the cost and three times the size of the E5 and its cousins, personally I would leave the E1 to the more serious SW dxers and go with the portable.