That's possible with cable and a VCR, but not with satellite because you can't just splice a satellite signal. I believe to do that you'd need two individual satellite receivers to pass along two distinct signals.
Believe me - I used to live on the DVR. But I certainly don't miss shuffling through tapes trying to figure out which tapes I could record on, finding the right spot on the tape, etc. A few years back my [now] wife started working retail, and would call all the time asking me to tape something for her. I couldn't keep track of what tape was what, what shows she had watched, where to line up the tapes, etc. I bit the bullet and bought a Tivo, honestly not expecting a life altering experience. It didn't take long to realize what we'd been missing. And she always leaves the technology to me, but one day I made her sit down to show her how to use it so she wouldn't have to make me come do it for her every time. Her only comment - "well that's easy!" And my son has been running the DVRs since he was 4 years old.
A DVR completely changes the way you watch TV. Not only does it simplify recording and watching more than you can possibly imagine, you wouldn't believe how often you would make use of even just the pause and rewind features. And by setting up season passes, you can automatically record your favorite shows. Even if you normally prefer to watch live vs. time shift, if you happen to miss a show for any reason, you've got it taped and waiting for you. You can even tape two live programs while watching a 3rd taped program.
And there is no additional charge for a dual tuner. You pay a DVR fee, no matter how many DVRs you have - I have 4 DVRs in our hose and pay the exact same DVR fee as I did when we had 1.
I understand the hesitation to move to something new, I really do. But I cannot recommend dumping your VCR for a DVR enough. You would not regret it.
I'm not sure I buy the line your provider is feeding you about dual tuners and buying your own - doesn't really make sense...