Your post wasn't what I expected from the subject line, but I'll get back to that.
As the others said, playing the chordal accompaniment more staccato usually helps. What is seldom said, though, is that learning to cut short the notes (lift the fingers off the buttons) of one hand while sustaining the notes (keeping the buttons depressed) in the other hand is not something that comes naturally to most of us. It can be learned, but it's easier said than done. It will take practice, and you may need to practice the two hands separately, then try to put them together without losing the difference.
Another thing to consider is not playing as many notes at once.
It's often possible to play less than a full chord, and yet give the listener the impression that the chord is there. E.g., with a tune in the key of G, playing just an F# and a C will normally "project" a D7 chord, even if neither the D nor the A is there. And any note that's in the melody doesn't have to be repeated (in a lower octave?) at the same time in the corresponding chord.
Don't play all the notes of the chord at once: You may already be doing the most common version of this, which is alternating the tonic (base) of the chord with the rest of its notes together. (E.g., for a G chord: low G, then the B and D above it, played together.) An arpeggio is another common variant, in which the individual notes of the chord are played sequentially, rather than all at once.
An "alternating bass" combines the two above ideas, just alternating the tonic with (usually) the fifth of a chord (G and D for a G chord, D and A for a D chord, etc.), and omitting the third. This is a common technique for guitar players (especially in the blues?), as well as being the "oom pah" of the tuba in the stereotype of a German brass band.
Then there are "harmonies" that aren't "chords". More about that, below.
I said I expected something different when I read your subject line. In fact, what I expected was a question about harmony "lines", rather than "chords". Coming to the concertina from a background of wind instruments (including voice), instruments that can play only one note at a time, to me the word "harmony" means a sequence of notes that contrast with the melody, not a "stacking" of notes into a chord. Some common examples are:
Parallel thirds: Along with each melody note is played the note two notes above it in the scale (musically known as a "third", because counting the melody note as "one", the harmony note is the "third" in ascending the scale). This is probably the simplest harmony to do on an English concertina, but on an anglo it won't put the melody and harmony into separate hands.
Parallel sixths: Musically, this is the same as harmony in thirds, but with the harmony notes then dropped an octave, so they sound below the melody. This kind of harmonizing can often be done on an anglo with the melody in the right hand and the harmony in the left. This can also be inverted, i.e., playing the melody in the left hand and the harmony in the right hand, giving a separation of an octave plus a third (a "tenth").
Parallel octaves: While some might object to calling it "harmony", playing the melody an octave lower (in the left hand) in parallel with the melody in the right hand does sound different from the melody in only one octave, and it can be quite effective.
Not strictly parallel: This is a catch-all for "everything else". A harmony could follow the melody rhythmically note for note, but take varying directions in terms of pitch. Or it could diverge rhythmically from the melody, with more than one note in either pairing with a single note in the other. And such harmonies can be multiplied, with more than two notes sounding at once, though not necessarily built upon a formal "chord" structure.
Bass runs: A particularly common example of "not strictly parallel". On any instrument that can play both chords and single notes, this technique is often used for moving from one chord to another in a series of steps, rather than a large jump. It can also add variety to the rhythm and motion of a piece, e.g., as a temporary change from a steady "boom-chuck" chordal accompaniment.
These are just a few simple suggestions that you can experiment with, if you have the desire, and they can be elaborated and combined in countless variations.
First of all, thanks to you all for your help. Playing the chords for a shorter time than the melody notes is what I'll try. It sounds like that is what I'm doing wrong - in addition to my hands working at different speeds. I guess I always thought that a chord was to be held for the same amount of time as the melody note. I've never had any musical training, by the way, so my terminology is not very accurate, I'm afraid.
I don't usually play the full chords either - only two notes which, I guess, is an interval rather than a chord.
I may not be good, but I'm having a grand time and learning something. That's the important thing.