20 Amp single receptacle on two 15 amp breakers?
What a mess...
Assuming that your diagnosis is correct - two wires in the same cable connected to totally separate breakers, you have several different problems going on:

A 30A dryer receptacle should be on a pair of 30A breakers. This is 2 independent 15A breakers. You don't add (though it sounds logical) 15A + 15A to get 30A, because the 30A refers to 30A @ 240V. Each of those 15A breakers is actually 15A @ 120V. The two existing breakers together gives you 15A @ 240V.
Breakers for a dryer or similar 240V load need to be in a pair that trip at the same time. That is important for safety. As you already discovered, you could turn off one breaker but still have power in the box. If you had a dryer plugged in and it tripped one breaker but not the other, you might think there was no power in the box and get zapped while trying to fix it. The only way to do that properly is with a pair of breakers that are side by side and connected together (really &quot;one device&quot;).
Most of your breakers are &quot;double stuff&quot;/&quot;half size&quot;. That is not in and of itself a problem. However, a pair of breakers to provide 240V needs to be either 2 full size breakers side-by-side or the middle set of 2 pairs of double stuff breakers, as I believe is the case with the 40A pair in the lower left.
A dryer normally needs two hots + neutral + ground. There is a grandfathered way of two hots + neutral, no ground. Your cable just provides two hots + ground or hot + neutral + ground. My guess is that it was wired up using the ground wire as the neutral wire, since black &amp; white were definitely hots. That is another Very Bad Thing.

So what you had before was:

Wrong size breakers - 15A instead of 30A
Wrong arrangement of breakers - not next to each other
Not common trip
Neutral on the ground wire

Basically, a disaster waiting to happen.
On to your new setup. You did not indicate if this is a 20A 120V device or 20A 240V device. My guess is that it is 20A 120V. If that is the case then:

You need to determine the size of the cable wires. My guess is either 12 AWG or 10 AWG. 10 AWG is what you need for 30A. 12 AWG is the minimum for 20A. 14 AWG is the minimum for 15A. If it is 14 AWG then forget about it and start over.
You need a single 20A breaker. You should be able to replace one of the existing double stuff breakers. You can replace the 15/15 with a 20/15 or the 15/20 with a 20/20. Do not upsize anything else at the same time - i.e., if you replace the 15/15, you can't put in a 20/20 because you have to assume that at least some of the wire on the other 15 is only 14 AWG.
In the panel: The black wire from the cable goes to the new breaker. The white wire goes to the neutral bar (except...GFCI, see below). The ground wire might be going to the neutral bar or the ground bar right now, or (hopefully only if this is the main panel) neutrals and grounds mixed together. But seeing what a mess you already have, anything is possible.
This will leave you with one unused 15A breaker, which can be used for another new circuit.

GFCI could complicate things here. Generally speaking, new receptacles in a garage (among other places) need to have GFCI protection. That can be at the panel, except not normally as part of double stuff breakers, and you are out of space to do anything except double stuff without a lot of extra work. So that means taking back the single plain receptacle and getting a GFCI receptacle (single or double) instead.
But I'd be really concerned about other problems. Knowing about problems with just one dryer circuit, who knows what else is possibly wrong. Touching &quot;anything&quot; in the panel could be a recipe for disaster.