Dropsy was heart failure: oedema of the legs, decreased walk distance, shortness of breath. Similar things would happen if you had malnutrition or endocrine disorders. Donne is observing the thirst of such, and using it as a metaphor for his desire for God: as if he was without water.
He is also referring to the death of earthly love: his wife is in heaven, and he will not meet him til he leaves this world. There are songs about this.
Donne refers to a love greater than this. In the end, all things lead to Christ, and Christ to God.
Since she whom I loved hath paid her last debt
To Nature, and to hers, and my good is dead,
And her soul early into heaven ravishèd,
Wholly on heavenly things my mind is set.
Here the admiring her my mind did whet
To seek thee, God; so streams do show the head;
But though I have found thee, and thou my thirst hast fed,
A holy thirsty dropsy melts me yet.
But why should I beg more love, whenas thou
Dost woo my soul, for hers offering all thine:
And dost not only fear lest I allow
My love to saints and angels, things divine,
But in thy tender jealousy dost doubt
Lest the world, flesh, yea, devil put thee out.