6.3V desired at 600mA or 0.6A.
8.4V available (assume no sag, but that's unrealistic).
To be wasted: 8.4V-6.3V = 2.1V
Common 3-pin regulators are marginal with only 2V to work in. Plus the "8.4V" will surely sag. Worse: the "8.4V" is a meter average, it is probably bopping up to 9V and down below 7V.
Regulation is NOT actually needed. If you get close to 6.3V on normal wall voltage, it will be close-enough for any wall tolerable voltage. ("Tolerable" in the sense that incandescent lamps don't burn way-dim or too-bright short-life.)
IMHO, a resistor is fine and appropriate. It also adds a trace of soft-start.
Resistor should be 2.1V/0.6A or 3.5 ohms, 2.1V*0.6A= 1.26 Watts (double and round-up: use 3W part).
In fact the "8.4V" will sag. How much? Dunno. You could dummy-load it and find out. 8.4V/0.6A is 14 ohms 5 Watts. That's awkward. Radio Shed has 10 ohm 10 Watt, I'd toss one on there and see if it holds anything like 6 or 8 volts. If not disappointing, build with about 3 ohms tacked in series with the heaters, see what you get, adjust the 3 ohms down until near 6.3V at heater. The 1 ohm 1 Watt parts commonly used for big-amp bias-check would be suitable.
> I am using 470uf cap, but i don't know if this OK
Rule-o-thumb for medium voltages is 1,000uFd per one Ampere current. That says 600uFd.
This gives around 1V ripple, which may be OK for 25V or 100V supplies (or 6V relays and LEDs) but looms large for 6V "clean" DC.
A single stage of filtering will NOT make clean enough DC to route near sensitive input wiring. (And the buzz is worse than hum.) Vastly bigger capacitance will help only slowly... even 4,700uFd is not real-clean at 0.6A load.
OTOH a 2-stage filter is at least 4 times better and the added R is begging for a second cap to make a 2-stage filter.
To ballpark a value for the second cap, use R and figure what C would be needed to pass 100Hz (or 120Hz). R-C calculators are available online, but a Reactance Chart is quicker. I'll Attach one for your wall.
Find 100Hz at the bottom, and 3.5 ohms on the left. Go up to the intersection. Follow a diagonal up-left line until you find uhf numbers. Looks like something between 100uF and 1000uF. Perhaps 500uF. Now multiply by at least 10, so it will actually chop ripple, not just bunt it. 5,000uFd.
Experience says both caps could be the same. OTOH two 4,700uFd caps are big and costly, even at only 10V rating.
Go ahead and try 470uFd 3.5 ohms 470uFd. With very careful heater routing it may be clean enough. But I'd leave room for 2,200 or 4,700uFd caps in case it needs to be even cleaner.