Hello DaisyK and welcome to the forums!
If both your old Sony and JVC camcorders have Firewire ports on them, it is for a reason. They are mini DV cameras, and in effect the ONLY way of capturing DV is via Firewire. Some mini DV camcorders do in fact have mini USB ports, but they are NOT for capture of the mini DV but for capture of still photos taken with the camcorder. Some might allow some video to be captured, but at best this would be low quality streaming video...
No, the only way -- I repeat -- of capturing DV/AVI or even DVD-quality mpeg-2 -- is via Firewire. Capturing DV will give you, moreover, the very best and easiest SD video for editing as it is in effect the identical video to that on the mini DV tape. Capturing via Firewire direct to mpeg-2 can be a bit too demanding for some computers. However, I note you have not given us any of your computer details. This is the sort of situation why we ask new users to fill out such details in their Profile during registration.
So if your new computer does not have a Firewire port, I can only urge you to have one installed. They are very cheap and very easy to install if you have a spare slot on your motherboard.
I again wonder about your computer specifications because I suspect they might have something to do with the other aspect of your problem. When you say "I tried to render and burn a dvd [but] it was too much and could not finish the burn", I have the suspicion you did your editing in the editing module, but then jumped straight to the burning module (Share > Create Disc > DVD). If that was indeed the case, then your project file -- rather than video -- would have been opened automatically in the burning module timeline. That means it would have to be converted to DVD-compatible mpeg-2 as part of an already complicated burn process. If your computer is not a particularly powerful one, this often is the straw which breaks the camel's back. It would be better to finish your editing, then while still in the Editor, go to Share > Create Video File > DVD (if your project is less than an hour long). This will render your project into a new DVD-compatible mpeg-2. You then save the project and start a new one. Don't bother giving the new project a name -- the objective is just to clear the timeline of your old project. Then you choose Share > Create Disc > DVD. The burning module will then open, but this time with an empty timeline. You then manually insert your new mpeg-2, build your menu and burn. This time it should hopefully complete the burn successfully.
As a footnote, if your project is longer than an hour, you should choose Share > Create Video File > Custom, then in the dialogue box which appears, you will need to lower the bitrate from its default 8000 kbps. As a general rule of thumb, 6000 kbps will allow 90 minutes to be burned to a single layer disc in very good quality and 4000 kbps will allow 2 hours. However, at that rate, the quality will be no better than decent VHS tape. Using one of the highly compressed audio formats like Dolby AC-3 will also allow a few more minutes of video to be burnt to disc.
