You can definitely date in residency, at work or outside of it (which I'd highly recommend lol). I really didn't have time during intern year in part due to schedule demands and in part just due to me needing a lot of alone time to recover from work and not wanting to be making small talk with strangers on my one day off a week, but once I got into second and third year and fellowship, it was pretty easy to make time for a date or two a week during lighter blocks. Among my coresidents, one was dating pretty frequently, one was in a long term relationship for a while (that ended due to non-residency related reasons), a few got engaged and subsequently married to their long term partners, a few had kids, etc...there's plenty of time for your love life if you prioritize it.
Also, don't get hung up on finding someone in residency - I didn't find the right person in training, but past this summer I moved to a new town for my first attending job and immediately met someone wonderful. Obviously who knows where things will go from here but we've been together for several months now and are excited about the future. Good things and people can come along any time!