I hope these floppies are not copy protected, otherwise what you plan to do will probably fail.
If your main computer is a Desktop system as opposed to a Laptop, and you are comfortable with a screwdriver, you don't even need to buy a whole old computer, just a naked 5.25 drive for PC will do fine. If the floppy cable currently inside your PC doesn't have the right type of connector (edge connector) for 5.25 drives you will need one that does, but many floppy cables have both types of connectors. Turn off your computer at the power supply (or unplug it). Open your computer, remove the floppy cable from the 3.5" drive and plug it (or the replacement cable you got) into the 5.25" drive. Make sure to use a connector AFTER the "twist" in the cable, the connector(s) before the twist are for a second B: floppy drive, but modern motherboards usually support only one A: floppy. Find an unused standard "IDE hard disk style" 4-pin power plug coming from your power supply and plug it into your 5.25 drive to power it. No need to put the drive into the computer case, just set it on a non-metallic surface and make sure it can spin freely. Turn the computer back on with the case open, go into BIOS setup and change the floppy type to 1.2 MB or 360 KB, depending on what type of 5.25" drive it is.
Now you can use WinImage or another imaging software to make images of your old floppies and you can immediately try them out in DOSBox. When done, simply put it all back together again as it was before and change your BIOS floppy setting back to 1.44 MB.
(OK probably you knew... but it is really easy to do).