This may seem like a silly question, but what type of DVDs are these, PMB707? -R, +R, -RW or +RW? The first three won't play in other gear until finalized, so are not the best choice for your type of recording task (adding short clips over a long period of time). But +RW does not require finalizing and will usually play in any current player, video console or computer. If you haven't yet tried playing the discs on other gear, test a few PCs, disc players and game consoles. Unfinalized Sony DVDs might perhaps play in a Sony PlayStation? PS2 and PS3 are odd ducks with surprising capabilities.
The Sony RDR-GX300 is an old model- really really old. Released in 2004 old. That vintage of recorders was the tail end of the first wave: most were very expensive and very flakey. Most Sony owners gave up on their GX300 and HX900 machines a long time ago: failing to play or record any disc was a very common problem once they reached their second birthday (never mind eighth). So it would probably not be a good idea to spend serious money for a used GX300 on eBay: the overwhelming majority have failed already.
Often you can still finalize discs from old machines on any newer model of the same brand (Panasonic, Pioneer, Phillips, Magnavox). But several big brand names stopped making their own recorders after the first couple years, opting instead to have other mfrs re-label recorders for them, thus leaving unfinalized discs made with older models stranded. Sony has done this at least twice since 2006, with no consistency among models: some were "real" Sony, some are hybrid Sony/Pioneer, and the most recent are usually rebranded Samsungs. Finalizing compatibility varies among these.
What you need is a second-hand Sony that still works and uses the older original Sony file system for dvds. Look on eBay or Craigs List for models made 2006 or earlier, but avoid any Sony model number that ends in "00" like your GX300: these have a "cult" following that inflates used prices beyond a safe risk. Also avoid the RDR-HX models for the same reason: very pricey used, and most won't finalize your GX300 dvds anyway. Some cheaper used Sonys that should be able to finalize your discs would be RDR-GX315, RDR-GX330, RDR-VX515, RDR-VX530 etc. If you have a friendly chain store nearby with current Sony recorders on display, you could bring one of your discs and ask them to let you try playing it in the display model. If it accepts the disc, buy that unit: its compatible. Your chances of this working are 50/50 at best: current models don't use the same file system, but if Sony asked Samsung to add backwards compatibility it might be possible. Doesn't hurt to try.
If you can't find a compatible, affordable used Sony recorder you might try PC software like ISObuster. The free trial version doesn't do much but the paid version is known to pull video clips off unfinalized dvds. You need to have some technical knowledge and skills to do this: the salvaged videos have to be "re-authored" using yet more software and results are not guaranteed. Best course of action may be to sit tight and wait for a good deal on an older Sony to pop up somewhere- you could try pawn shops, Goodwill, etc.