The modern usage was beginning to come into the language in Shakespeare's time; in the Oxford English Dictionary, Shakespeare is the first recorded user of the word in this sense, when he has King Richard talk about a mother "playing fondly with her tears" on meeting her child after an absence (Richard II, III.ii.9).
But usually Shakespeare uses the word in its original sense - "foolishly, stupidly". This is the sense needed when, in the same play, Fitzwater talks about "fondly" spurring on a horse (IV.i.72), where the meaning "lovingly" conflicts with what is involved in spurring. The linguistic context usually helps: "What my great-grandfather and grandsire got My careless father fondly gave away", says Clifford, imagining what Prince Edward might say (Henry VI Part 3, II.ii.38): here "careless" immediately suggests the required meaning. But in a sentence like Adriana's to Dromio of Syracuse, "How fondly dost thou reason" (The Comedy of Errors, IV.ii.56), the actor must know the older meaning before she can say the line in the appropriate tone of voice.