Hi OP. Sounds like a tricky situation. Here's what I would think about doing:
Acknowledging and validating his sad feelings. Massively, massively, massively. Recognising them, validating them, referring to them. Not minimising them in any way -
no mention of why real life means his dad has to be away or his mum has to work a long day. Just lots and lots of 'I can see that you are feeling sad about...that must feel awful for you' etc. See book How To Talk So Children Will Listen and Listen So Children Will Talk for this method. It's fantastic!
Try to clarify with your DS what is making him feel sad about being separated from his parents. Lots and lots of listening from you. Is it his dad being away so much? Is it his mum's job? It could be one of these, or both or something related but different. You need to know what it is. He needs to be able to explain it without you chipping in and coming up with all the extenuating circs as to why things are the way they are.
Then I'd try to drill down more about the specifics. For example if it was his dad being away I would ask him what specifically makes him sad. Is it that he's not home enough? Is it that he doesn't know when he's going to be home? Is it that he doesn't feel connected enough to his dad when his dad's away? Is it that he doesn't 'get' enough time with him when he is home? Etc.
Then I would ask him to have a think about what would make this sad situation a little better for him. Give him a couple of days to ponder. Make a list of everything he says, whether impossible or not. Add in some suggestions of your own. Pin the list up where he can see it. Identify together some from the list that are doable and do them. Keep the list in view as a reminder for both him and you of what he needs to change. Keep referring to it like a working document so your son knows it's a short/ medium term priority.
Practically I would hugely cut down on outside activities: swimming, rugby, sleepovers etc. He's at young end of scale for them and at mo it's time with Mum and/or Dad he needs. Plenty of time for the activities when he's a bit older/ less anxious.
I don't know what outcome of your son talking about what is making him sad will be - what the root of it is - but I think in medium-term you and your husband might need to acknowledge that both your working arrangements together may need to be re-evaluated. And consider working less/ differently. Not easy to acknowledge but may be necessary.
You are so right that you are both parents and partners to each other and it's very easy to forget the partner bit. But as you said, the children are your priority and I think right now things like date night/ kids' club need to go on back burner whilst you try to make your son feel more secure. This won't be forever, you don't need to martyr yourself on a bonfire. It's just while your son is struggling.
I wish you luck in sorting it out. You have a very busy life and you are having to do an awful lot - it's not easy.