OK. I wasn't aware you'd already upgraded to 1803. My own TSSTcorp DVD+/-RW SU208GB is working just fine in 1803. But wait a minute, you said the driver was...
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Current driver is: C:\Windows\system32\DRIVERS\cdrom.sys
Provider: Microsoft
Driver Date: 6/21/2006
Driver version: 10.0.16299.15 (WinBuild 160101.0800)
It seems very unusual that the driver is so old? And, I was curious about it being a 32-bit driver?
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16299 is 1709. Here is 1803 (though apparently the date is unchanged)…
What makes you think you upgraded? (But it really should work in 1709 too). Let us consider this...
How to: Perform a Repair Upgrade Using the Windows 10 ISO file
That can be done with an ISO or with Installation Media. In all cases, you boot to Windows, open the media in File Explorer (or click the ISO) & click Setup.exe. (It ran a bit over an hour to lift me to 1803. It did create a Windows.old - but I'm unsure it was full-fledged, didn't seem big enough when I first looked at it.)
But - I wonder - what will it do, if you've already got a C:\Windows.old? Do you have one now? Will it even seek to create one, if you are doing a repair (over-installing the same version) instead of an upgrade? Normally, that holds the prior OS version & normally will be deleted by Windows in as little as 10 days after the upgrade. Or it can be deleted with Disk Cleanup.