Mrrzy
I have known two really good "ear" piano players, and neither of them started very early in life. One of them was a fellow whe went by the name of "Goody" Goodman. I met him because he had a really cute daughter who used to come to our neighborhood square dances. He had his own band, and played a regular schedule of local gigs - including High School dances, which is mostly where I got to hear him. I suppose he was probably in his early fifties when I knew him, and the only thing I know is that he once said that he didn't start playing the piano until he got out of the army (WWII, I presumed). The amazing thing is that he was good, and he did it all with only three fingers on his left hand. (Not an uncommon affliction then, given the number of machine shop & sheetmetal workers in our area.)
The second good play-by-ear piano banger I knew was the brother of a close friend while I was in Jr Hi & H.S. He claimed that he learned to play while he was in the Army. We liked to kid him about being a "combat vehicle operator" in Korea. His vehicle was a typewriter, and he said he "spent a lot of time in the dayroom, where there was a piano, and just fooled with it until it started to make sense." He could not read music, and played entirely by ear. While his preference was for "churchy" stuff, he obviously had learned enough other kinds of music to avoid complaints from those with whom he shared his dayroom. Pretty good ragtime, boogie, and the then-popular big-band swing numbers.
While neither of these guys may have been as old as you are now, they both apparently learned when they were well past the age when a lot of people start.
Since you say that you want to sing with your music, that pretty well rules out an instrument that you have to stick in your mouth. While in my younger days I could demonstrate that you can sip your beer while simultaneously playing the pennywhistle AND not dropping the cigarette out of the other side of your mouth, its not a recommended practice, and that kind of "trick" stuff does seem to get harder as one mellows.
The mountain dulcimer is probably the simplest instrument that you can sing along with. Beginners usually play on one string (or one pair - tuned alike, a double course). Frets make a diatonic scale, and most beginners push the string against the frets using a "noter" that usually looks like a short piece of wooden dowel. Two other strings are tuned to make a chord with either the open 'melody' string, or with the fourth or fifth of the 'melody' string. By strumming across all of the strings, you can get a sort of a chord background, or you can use them as "drones." On most 'modern' dulcimers, one extra fret (at the 6.5th note of the scale) is added to the basic diatonic set of notes. This lets you play in either of two scales. You can, and most dulcimer players do, retune the strings to another key when needed. More advanced players will usually use their fingers to press the strings against the frets, and this allows a person to actually play a fairly complex chord background.
If you are a person who "thinks melody" primarily, the mountain dulcimer would be a very good, relatively cheap instrument to get you started. Its main weakness is the diatonic scale, which slightly limits what songs are easy to play, and to a certain extent makes it mainly a "melody" instrument, especially for the beginner. There is also the fact that it is not (usually) very loud. Until you are ready to graduate to pickups, mics, and such, it should likely be considered an instrument for the "solo" performer.
If you mainly want to play melodies, a logical step up in complexity and instrument versatility would be a mandolin or 4-string (fretted) banjo. In this context, the banjo is to be played in 'irish' style, as a melody instrument - not the popular (and of course beautiful) bluegrass style. Either of these is fully chromatic, so you can fairly easily play almost anything on either instrument. The banjo has the advantage that the strings don't chew up your fingers quite as much; but you should consider whether it may be too loud for your singing voice - and you have to put up with all those jokes!
A general rule for accompaniment of singers is that the accompanying instrument should NEVER play the melody while it is being sung. You can get by with violating this rule if you are accompanying yourself, but the rule is probably one of the reasons why guitars are so popular with singers.
If you naturally can "think in chords," then you would probably be better off with an autoharp or guitar. The beginners learning curve is probably a little steeper on the guitar - you can probably pick up simple strums more quickly with the autoharp (just an opinion - please hold the flak!).
The main thing is to get something you like, try it out, and go from there. Whatever you get, keep it close by, pick it up frequently, play with it, fondle it, love it, but DON'T try to learn everything in one session. You will hurt yourself, no matter what instrument you choose.
I read music but a lot of good players don't. I find that following the dots is the only way I can play a new piece the same way twice, and it greatly speeds my learning process. Once I feel I've learned it though, I don't worry about playing it exactly the same every time. It's more fun to "put some of yourself" into it. And I hardly ever look at music when I'm playing with others, even for unfamiliar tunes.
By all means learn to read music - but don't worry about making it a crusade. It's just a tool for making life easier - and more fun.
Pleasant tones.
John