The power system design is terribly important - currently you will have huge variations in voltage and drain the batteries ultra fast, the whole time reducing the voltage of the servo. You will never get really consistent speeds/behaviour. And it will be super heavy and annoying.
I suggest you get a 7.4V 3-4 amp-hour lithium ion or NiMh rechargeable battery pack, whack a 5Amp fuse in series, and then either get 4 separate 5V 1A linear regulators (with low "drop out"! this is very important) using nice big TO-220 package 3-pin parts for heat dissipation reasons. Always put large bulk capacitors on the output of these as well, such as 470uF -> 1000uF with at least 6.3V ratings, but ~16V would be better in case the regulators fail as shorts to the 7.4V rail (otherwise you'll have explosive 6.3V rated capacitors haha!)
The input capacitors to those linear regulators need to be less crazy, a 16V 10uF ->22uF will work fine.
The alternative to the linear regulators, and which will be more expensive and much harder to find, is to get a switchmode supply that can provide around 4-5 Amps continuous (pulsed currents can be much higher, usually). You may find something handy on Pololu's website, which caters for these sorts of applications. In this case you would want a fixed 5V output or adjustable output which you can make to 6V yourself, using a multimeter to check the output voltage. The name of the step-down DC-DC voltage converters you would want to search for are "buck" converters. "Boost" converters are ones that step voltage up.
Also, try to reduce the length of Ground wires connecting things together. If you have long lengths of thin ground wire, you may have differences in ground potential (wires are resistors!). The "power" ground loop though is the most important, between the battery, power systems, and the servos. The input signal and common ground between the Arduino and the servos is okay to be thin.
Finally, why not use a wall power adapter? If the robot arm is not mobile, then surely you can put a wall-plug based supply somewhere and get rid of all the batteries.
As for your worries about the current-sharing of the servos, this is why it is important to either have separate regulators, or have a big one with good current output. The capacitors I mentioned are also important to provide high transient currents as the servos turn on.