If my servo can be run from the pin, (someone said it could and someone said it couldn't)
Some can run from an Uno pin. Yours.....
As for 3.3V, check the device specs.
it's voltage specs are 4.8v-6v and its min and max current are respectively 5ma and 700mA.
Uno pin can source 25mA continuous safely. You would put a resistor between the pin and the servo to prevent more current from being drawn from the pin.
So if the board can only output 150mA obviously there's a problem there, since the servo can need a lot more.
The servo can use more, which your project may or may not need.
Is there a way I can bypass/negate whatever controller the board is using that limits the current? Or is it a better idea of doing the FET Driver like this one: https://tinyurl.com/ya8a8lxz Would that particular FET Driver be capable of driving both servos without using up all the board's current/power?
You can run a power wire from what supply you have to a transistor (FETs are very efficient, BJTs are very cheap) that the controller turns on and off to let the power flow to the servo. The servo ground wire should be connected back to the power supply (does not have to be the same as board power but those grounds should be connected so that board ground = servo ground). This will let you use 5V or 6V to run the servo and 3.3V for the board.
The important part is that motive power can run outside of the controller while the controller can switch that on and off using small power.
You will need to spend time learning about transistors but for this there are schematics you can follow. Which transistors you get will determine what schematic but you can get plenty of help with that right here.
A couple of years ago I got $10 worth of IRLZ44N's on eBay, 60 of them at 14.5 cents ea. They can switch > 10A, more than enough for my needs. For just a few (like 10) they were about 35 cents ea and some sellers wanted close to $1 for 1, it pays to shop around! You may find better deals, you may find other FETs (field effect transistor) you like more. The IRLZ44N (and all that end with N instead of P) is best used to drain power from a circuit to ground (the power is always there but can't flow until the path to ground is open through the FET) and that's what affects the circuit you use.
Heck, I have relays that control with 3.3V or 5V and can switch 10A of house power (120VAC here), those are cheap ones too. The thing about relays is that they can't take being switched on-off-on-off quickly for long, it will make them hot. Transistors can be switched very quickly though with FETs only full ON and full OFF or they will get hot.
I hope that the background will help you get into this easier.
And last tip; look into DC-DC converters. Cheap ones can output 1.5A continuously.
You have loads of options.