I do not have adopted DC but re rewards for being good - can you set up a "token" system. Set out say 5 or 6 expected behaviours throughout the day, make them fairly wide reaching but addressing issues you have. The expectations need to be reasonable for his development and personality, but you're not aiming for/expecting him to "win" all of them every day, not at first.
e.g. ours are:
Timeliness - not making us late, getting ready in plenty of time, not kicking off when it's time to leave, listening to reminders about you have X minutes left, not getting distracted/coming back when it's pointed out you've got distracted.
Respect for people - using nice/kind voice/words to ask for stuff, not being rude or stroppy or attitudey, no name calling, no violence, acknowledging people/requests (I don't expect immediate compliance).
Respect for privacy - doors closed when using the bathroom/dressing, don't barge into bedrooms, knock first, no inappropriate touching, stop when somebody asks you to when playing physical games, etc.
Being helpful - completing a list of (simple) chores, not obstructing people trying to get on with stuff.
Eating nicely - sit nicely at table (not perfectly, but bum on seat facing the table, not getting up randomly etc)
Keeping clean - self care stuff (shower/bath when appropriate - we have a board for which days this happens), teeth twice a day, hands/face washed when appropriate, not leaving massive messes in wake - obv normal kid stuff is fine, and not expected to be done without reminders.
You could add/swap out stuff like:
Respect for property
Respect for resources (not wasting stuff)
Respect for learning (doing homework, not disrupting any lessons)
Controlling temper (dealing with anger constructively)
Telling the truth
Don't make the list of expected behaviours too long - 5 or 6 seems about right. Don't give him the list of "This is what not to do!" specifically but talk about what you do want to see - what's why all of the titles/categories are positive morals/behaviours/etc. Don't particularly dangle/threaten loss of tokens throughout the day, but use the words. "That really helped us to be on time, well done!" "That isn't very respectful. Would you like to start again and try again?" "X, you are not respecting my privacy right now" "Thank you, that's really helpful." (etc). So you can pass comment on behaviours but don't go "Right that's it you've lost your token!" or "Do you want to get a token or not??" At the end of the day, run down the list and allocate which ones he achieved. Tell him which ones he's getting. "You got three tokens today, for being helpful, respecting privacy and being on time." Don't comment on the others, but if he asks, then you can say "Well you lost the respect token because your teacher told us that you shouted at her today." You can talk about it, but follow his lead - don't launch into a lecture especially just before bed. The best part about the token system - you can remind him - is that he always has another chance the next day. He always gets to start again and try again.
Then you can also add "instant" tokens for particular flashpoint behaviours or achievements you want to work on (we have one for trying something new and one if all chores are completed before a certain time) or just to reward something immediately if you see that he's done something well, something hard or something big for him.
Obv you still have to use ongoing behaviour management such as if he is trying to hurt you/himself then you must prevent that. And for bigger transgressions you might need to do a consequence or some problem solving or whatever constructive approach is helpful. But this does allow you to ignore some of the more minor stuff while still addressing it by encouraging the opposite.
Make the tokens actual physical things - plastic coins/chips or make some out of card and sticky backed plastic so he can have them and hold them and be given them immediately. Let him save them up over time or spend them immediately on rewards - something specific (screen time in our house) to a range of stuff (from monetary "things" to experiences, privileges or other treats). Consider having some kind of bonus if he amasses eight or ten at once. I'd recommend having one of the rewards be something he can do for bedtime so he has a chance to spend them immediately on receiving the bulk of them for the day.
It's nice, because it focuses in on the behaviour you want and clarifies that, (both for you and him), forces you to pick out the most important issues, and allows you to focus on what he has done right. You should be able to award at least one or two even on a bad day, if you can't, or you're regularly getting v low numbers (4-7 should be typical taking into account all-day and instant together) then it might be that the expectations are too high and you need to take them down. Set him up for success - you can increase expectations later.