Wow zombie that's a shocker. I think that first and foremost you need to remember that both of you need sleep, not just you but your DS to develop, be rested and happy.
Daytime routine is important - an overtired baby is more difficult to get to sleep and more wakeful when they are asleep so you need to look to achieve 2 good daytime naps. I would also suggest moving to a by-the-clock routine. Stick with it for a week to see if he adapts. It will mostly feel like it isn't working to begin with but after a while, they usually get the idea.
So;
7am - get up and dressed if not awake already
7.30am - milk
8/8.30 breakfast
11am lunch
11.30 - 1 nap
2pm milk
4pm dinner
4.30 - 5.30 nap
6pm supper
6.30 bath if required
6.45 IN THIS ORDER - pjs, gro bag, milk, book, upstairs, cuddle and bed
Putting to sleep as follows;
Nap time routine - nappy change, gro bag, bedroom, curtains drawn, the same book and then bed
Put him down wide awake and leave him to it for 5 minutes. Go downstairs and keep yourself busy tidying the kitchen etc. Don't hover outside his door. When you go back, listen before you walk in. If he's wailing then go in. Don't take him out of his cot, just say, 'it's okay, it's just time to go to sleep now.' A quick shhh and a rub on the bum and then leave. Settling isn't the purpose of the visit, it's just to let them know you're there.
Then back downstairs for 10 minutes and repeat every 10 minutes. Don't stay for too long and go in too often - it just winds them up if you do that IMO.
If you don't want to leave him to cry, you can do gradual withdrawal where you put them down wide awake but then sit in the room, not making eye contact. This is obviously harder as you have to listen to the crying but you may feel better about not leaving him.
Night wakings;
Upon waking, always give him 5 minutes before running in
At 10 months I wouldn't be offering milk from 7-7, maybe a dreamfeed if he isn't on many solids
Treat night wakings as above - either leaving in intervals or gradual withdrawal
I hope this helps. The biggest thing here is consistency.