A full wave bridge rectifier and a filtering capacitor should work towards eliminating the flicker when powered via an AC source.
I'm not an "Electronics Kinda' Guy". Could you please show me what these items you speak of look like and where I may find them? Will using these diminish the brightness of the LED's?
The bridge rectifier is a 4-legged device, the capacitor is a 2-legged device. The above are representative listings from eBay showing you'll have leftover parts to share. If you go this approach, one of us will give you exact hookup instructions. The brightness is controlled by the size/value of the resistor. It appears you entered 16V to the LED-calculator program; these programs assume smooth DC voltage in calculating the resistor value. If you actually supplied 16V AC, you actually getting less brightness than you think; the math is a somewhat tedious but do not be concerned about brightness. Using the approach suggested, you will eliminate the flicker AND have at least as much brightness available based on choice of resistor value.
Separately, you have the 1033 available so it's "free" but if you don't want to mess with bridge rectifiers and capacitors, I like Leo's suggestion about finding a DC wall-wart that puts out a fixed DC voltage. Then all you need to mess with is the resistor value. This would be a good approach if, say, you need to fit all the components on the tower itself. Or if you plan to do more LED lighting of buildings, station platforms, streetlights, etc. that would of course also require DC voltage.
In either case, I see you are messing with resistors that have power ratings of 1 Watt or more. Apparently you have some 22 ohm, 1 Watt resistors in hand for your initial try. I see that on eBay you buy a pack of 20 1-Watt resistors for about 5 cents each. In other words, you don't have a variety of 1-Watt (or 2-Watt) resistors lying around! The point is if all you have are a bunch of 22 ohms resistors, and you want to experiment with brightness, you can "make" a 11 ohm or 44 ohm resistor using two 22 ohm resistors. And by combining 3 or 4 of them, you can "make" any resistor value relevant to your floodlight application. Then, once you find a suitable brightness, you can choose to buy a single resistor of the chosen value...or just neatly bundle up what you have!