I'm really lucky that Z usually takes my concerns really seriously and changes his behavior accordingly. I hope your partner does too because that in itself is a way we can feel really special.
We had the gaming issues as well in the beginning of our relationship. It was a huge shock for me when we moved in together and Z didn't seem to want to come to bed with me at night, preferring to stay up playing computer games until the early hours. I actually left over it. I said that I didn't see how he could love me in the way I wanted if he would rather be on the computer most nights than in bed with me.
Then it was him who was shocked. He had no idea that I equated computer time at night with lack of love. He had been in such a bad marriage previously that they just didn't do anything together and he had no real idea of how a good relationship operates. To save money he and his ex-wife would always go out separately so as not to have to pay babysitters. It's taken me over two years but he is now an excellent partner who loves coming to bed with me and going on "dates", as he calls it.