You can create multiple profiles to break up your backup tasks into smaller segments, if you like. For each profile, you can specify which folders and files to backup.

Outside of these somewhat hard-to-find settings, everything else should be relatively obvious. Now, enable the 'Reset the archive file attribute on files once they have been copied' option for the profile. Next, go to the profile's Copy/Delete tab, and then the Advanced sub-tab (the sub-tabs are along the top of the profile's setup window). In that tab, check the checkbox for 'Only copy files that do have the archive attribute set'. To accomplish this, create a new SyncBack profile and go to the Compare Options tab in the profile's setup window (the tabs are positioned vertically along the left edge of the profile's setup window). The next step will be to create a profile in any of the SyncBack products to only copy files with the Archive attribute enabled, and to remove the attribute once the copy operation is done. The first step will be to use any tool you like (such as attrib, which is included with Windows) to enable the Archive bit on every file on your drive partition(s). I have used their software for many years and find it to be of high quality. SyncBackFree is gratis, SyncBackSE is within your price range, and SyncBackPro is at the high end of your price range. You can compare the different editions here. You accomplish your goal with SyncBackFree (portable version available here), SyncBackSE, and SyncBackPro.

Syncbackfree portable