I was in this situation. A few years ago my husband told me how much he was struggling with our lack of sex life. I hadn't realised how much our sexual connection impacted on his emotional happiness and that sex was his primary way of receiving and showing love. Once I understood that it wasnt just a case of a 'horny husband' I took his feelings seriously.
I have a very low sex drive, always have done. I always had an excuse to avoid sex. Once I understood the damage that our lack of sex was doing to our relationship, my mindset started to change.
I decided to try to always be open to sex. If I just went with the flow, I often ended up enjoying it. If I wasn't getting much out of it, I would just take it as an opportunity to be affectionate and loving with my husband. I didn't lie there like a sack of potatoes, and nor did I fake anything. I was still able to make it a special and close time. Sometimes if I really dont want to, I will either do something 'alternative' or rearrange for the morning (and do it).
We have sex Once a week and try to never let a week go past without it. I could easily go months without it and not miss it but I decided that by investing in our sex life, I would ultimately be investing in our family. I never regret saying 'yes' to sex with my husband, whether I truly get into it or not. I don't even care what anyone thinks of my approach: I love my husband, I love our family and don't want to split up or for him to be unhappy in our marriage.
I always try to be affectionate outside of the bedroom too. Hugs, kisses, little moments together.
OP, as an aside: I once read a problem page where a sex therapist answered questions and a reader wrote in to ask how she could reconnect with her husband...long talks? A date night? Join a club together?
The therapist said 'just have sex' and I've found it true in my case.
I also used to worry that I was really boring in bed and that my husband really wanted some kind of kinky woman of the night to thrill him in unique ways each time. I read a kind of parody about this, which said that men are just happy to have any kind of sex and I realised its true! I don't have to be hanging from the rafters. I just need to be me, to be there, to share that connection.
Over time, I have seen how much happier we both are. I even initiate it sometimes because I enjoy the closeness rather than the sex itself. If my husband lost his libido and didn't want to be sex anymore, our relationship would suffer even though I don't have much of a sex drive and don't get much out of it physically.
Not everyone will agree with my approach but it's not their marriage. I always put the family first and try to find a solution from that vantage point.
It sounds like your marriage has a lot of potential for things to improve. Hope all goes well.