Have an experienced uke player try your uke... he or she might recognize the problem instantly and help find a solution.
Here are possibilities:
(1) Like you suggested, fret more firmly. Place your fingers close, but not on top of, the frets. Allow the string to be firm against the fret with a clear vibration to the saddle/bridge. Often I see players of fretted instruments attempting to fret the string midway between frets. Move your fingertip up as close to the fret as possible without touching the fret.
(2) Strum lighter. Hard strumming will cause the strings to vibrate wildly, perhaps striking random frets between where the sting is fretted for intonation and the saddle/bridge.
(3) Raise the saddle. I changed strings on a fine traveler tenor uke attempting to rid the uke of minor fret buzz. But no matter my technique, many strings, many frets buzzed. The strings were too close to the frets. Any vibration of them at all would cause them to touch random frets. I had no new saddle material to make a slightly higher saddle, so I tried various small sections of uke strings under the saddle to raise it. The saddle was cut so low it took an Aquila C string to raise it enough to rid the uke of fret buzz. I took the uke back to the seller and he cut a new saddle exactly the thickness of my C string higher and now it never buzzes.
(4) A few ukes have truss rods in their necks. If the neck bends back (anti-relief), loosen the truss rod to allow the strings to pull the neck to allow a little relief. Look for Google images under guitar repair to see how truss rod adjustments affect the flatness and curve of a fretboard. This is a job for an experienced luthier, and should not be tried by a beginner on an expensive instrument.
(5) Thicker (thus tighter) strings. Material matters less than thickness. A thicker string sounds lower in pitch than a thinner string at the same tension. Therefore, thicker strings need to be tighter to be at the same pitch as a thinner string in the same spot on the same instrument. Use thicker, tighter strings to lower the action (the distance from the top of the frets to the string) on any string instrument. Thinner string have less tension so they are easier to press down, but their looseness also causes them to vibrate farther, causing more fret buzz at lower action.