The ColorEdge CG243W is a 1902 by 1080 display. It has a DisplayPort input as well as its (Dual-Link) DVI input. At less than 1920 wide, it does not actually need Dual-Link DVI, and would also work fine with a single-link DVI adapter instead.
The ColorEdge CG277 is a 2560 by 1440 display. It has DisplayPort as well as HDMI and Dual-Link DVI. Since it is wider than 1920 wide, it can Not be be run on Single-Link DVI, and results under HDMI may be disappointing (such as only 30Hz refresh).
If you have a certified high-speed HDMI cable (look behind your TV set) you could try borrowing that cable for an experiment. HDMI was invented for consumer TV sets, so at higher resolutions, it tends to be difficult to stabilize with the desired refresh rates. Your Mac Pro 2013 dark cylinder has an older version of HDMI output.
The best results are obtained by using DisplayPort family cables, when possible. ThunderBolt-display and USB-C display are close cousins of DisplayPort.
The Mac Pro 2013 has six ThunderBolt-2 ports. When you plug in a Mini DisplayPort cable (such as the input to the Apple Dual-Link DVI adapter) it picks up Mini DisplayPort subset directly. A cheap "all wires" adapter/cable from Mini DisplayPort to [full size] DisplayPort would allow you to test without the Apple Dual-Link DVI adapter and make progress more quickly.
Here is a high-quality cable that would work for either or both of your displays:
Monoprice Select Series Mini DisplayPort 1.2 to DisplayPort 1.2 Cable, 6ft - 13373 $5.99
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