Am I missing something? It has come up that my partner and I only have one card left in our hand and we haven't make a Canasta so we can't go out. So we end up picking up and discarding in an endless loop. Is there any recourse other than hoping either me and my partner will pick up a 6th or 7th card to make a Canasta and go out? In the meantime our opponents are racking up points left and right.
Assuming that the discard pile is not frozen; then you can continue to score points if your opponents ever discard something that you can play. This will also limit what your opponents can do to score points; they wouldn't be able to just run through the whole deck and score. However, you are still in a bad position; both you and your partner made a major mistake by going down to only one card in hand wihtout having a Canasta yet.
If your opponents are smart, they will take advantage of thisby keeping the discard pile continually frozen; and then they can discard any card safely no matter what; and they will be able to pick up the entire discard pile by using cards in their hand when you discard something they have (which you'll be forced to do every few plays). This way, your opponents can literally score every card in the deck that you haven't already scored. This could even end the entire game in one hand.
There's no rules thing you are missing here; you simply made an error; and it was worsened by the fact that your opponent made the same error. Even though having more cards in hand when your opponents go out is a bad thing; in general having more cards in hand is a good thing. The more cards you have in hand, the more likely it is that you will be able to pick up the discard pile and score more points. If you are worried that your opponent is going to go out soon; you can play your higher-scoring cards to get them out of your hand, while still keeping a few lower scoring cards in hand as to not get stuck.
This also depends on how close you are to a Canasta. If you have a six card set already; then you may be able to go down to one card and hope that you draw the seventh card. This is still risky though. If you have nothing larger than a 5 card set; then you have literally made it so that you can never make a Canasta or go out; because you can't ever add 2 cards to it at the same time.
Yes, you missed the fact that getting down to one card, or just a low number of cards, is bad and will limit what you can do. Because of this you where not able to score points and your opponents where able to score a lot of points. I would also guess that because of the situation you had to discard a lot of cards that would help them out in scoring many more points while they could avoid discarding ones that would help you.