I found these definitions in an online version of Samuel Johnson's dictionary. I'm sure Shakespeare did use it in the sense that you mean (and I know you are much more an expert on this than I will ever be!) It seems like in the very first definition here, the Shakespeare example is playing with the meaning of the word--meaning common sensical but also capable of feeling.

However, I wonder why the very last entry suggest that the meaning of 'reasonable, judicious, wise' was of "low use" at that time, which was just a bit before JA?

SENSIBLE 1 adj. [sensible, French; sensilis, Latin.]
1. Having the power of perceiving by the senses.

Would your cambrick were as sensible as your finger, that you might leave pricking it for pity. Shakespeare.

These be those discourses of God, whose effects those that live witness in themselves; the sensible in their sensible natures, the reasonable in their reasonable souls. Raleigh.

A blind man conceives not colours, but under the notion of some other sensible faculty. Glanv. Sceps.
2. Perceptible by the senses.

By reason man attaineth unto the knowledge of things that are and are not sensible: it resteth, therefore, that we search how man attaineth unto the knowledge of such things unsensible as are to be known. Hooker.

Is this a dagger which I see before me, The handle tow'rd my hand? Come, let me clutch thee: I have thee not, and yet I see thee still: Art thou not, fatal vision, sensible To feeling as to sight? Shakesp. Macbeth.

The space left and acquired in every sensible moment in such slow progressions, is so inconsiderable, that it cannot possibly move the sense. Glanv. Sceps.

It is manifest that the heavens are void of all sensible resistance, and by consequence of all sensible matter. Newton.

The far greater part of men are no otherwise moved than by sense, and have neither leisure nor ability so far to improve their power of reflection, as to be capable of conceiving the divine perfections, without the assistance of sensible objects. Rogers's Sermons.

Air is sensible to the touch by its motion, and by its resistance to bodies moved in it. Arbuthnot on Air.
3. Perceived by the mind.

Idleness was punished by so many stripes in publick, and the disgrace was more sensible than the pain. Temple.
4. Perceiving by either mind or senses; having perception by the mind or senses.

This must needs remove The sensible of pain. Milton.

I saw you in the East at your first arising: I was as soon sensible as any of that light, when just shooting out, and beginning to travel upwards to the meridian. Dryden.

I do not say there is no soul in man, because he is not sensible of it in his sleep; but I do say, he cannot think at any time, waking or sleeping, without being sensible of it. Locke.

The versification is as beautiful as the description complete; every ear must be sensible of it. Broome's Notes on the Odyss.
5. Having moral perception; having the quality of being affected by moral good or ill.

If thou wert sensible of courtesy, I should not make so great a shew of zeal. Shakespeare.
6. Having quick intellectual feeling; being easily or strongly affected.

Even I, the bold, the sensible of wrong, Restrain'd by shame, was forc'd to hold my tongue. Dryd.
7. Convinced; persuaded. A low use.

They are very sensible that they had better have pushed their conquests on the other side of the Adriatick; for then their territories would have lain together. Addison.
8. In low conversation
it has sometimes the sense of reasonable; judicious; wise.

I have been tired with accounts from sensible men, furnished with matters of fact, which have happened within their own knowledge. Addison.
SENSIBLE 4 adj. [sensible, Fr. sensilis, Lat.]
1. Having the power of perceiving by the senses.

Would your cambrick were as sensible as your finger, that you might leave pricking it for pity. Shakespeare.

These be those discourses of God, whose effects those that live witness in themselves; the sensible in their sensible natures, the reasonable in their reasonable souls. Raleigh.

A blind man conceives not colours, but under the notion of some other sensible faculty. Glanville's Scepsis.
2. Perceptible by the senses.

By reason man attaineth unto the knowledge of things that are and are not sensible: it resteth, therefore, that we search how man attaineth unto the knowledge of such things unsensible as are to be known. Hooker.

Is this a dagger which I see before me, The handle tow'rd my hand? Come, let me clutch thee: I have thee not, and yet I see thee still: Art thou not, fatal vision, sensible To feeling as to sight? Shakespeare's Macbeth.

The space left and acquired in every sensible moment in such slow progressions, is so inconsiderable, that it cannot possibly move the sense. Glanville's Scepsis.

It is manifest that the heavens are void of all sensible resistance, and by consequence of all sensible matter. Newton.

The greater part of men are no otherwise moved than by sense, and have neither leisure nor ability so to improve their power of reflection, as to be capable of conceiving the divine perfections, without the assistance of sensible objects. Rogers.

Air is sensible to the touch by its motion, and by its resistance to bodies moved in it. Arbuthnot on Air.
3. Perceived by the mind.

Idleness was punished by so many stripes in publick, and the disgrace was more sensible than the pain. Temple.
4. Perceiving by either mind or senses; having perception by the mind or senses.

This must needs remove The sensible of pain. Milton.

I saw you in the East at your first arising: I was as soon sensible as any of that light, when just shooting out, and beginning to travel upwards to the meridian. Dryden.

I do not say there is no soul in man, because he is not sensible of it in his sleep; but I do say, he cannot think at any time, waking or sleeping, without being sensible of it. Locke.

The versification is as beautiful as the description complete; every ear must be sensible of it. Broome's Notes on the Odyssey.
5. Having moral perception; having the quality of being affected by moral good or ill.

If thou wert sensible of courtesy, I should not make so great a shew of zeal. Shakespeare.
6. Having quick intellectual feeling; being easily or strongly affected.

Even I, the bold, the sensible of wrong, Restrain'd by shame, was forc'd to hold my tongue. Dryden.
7. Convinced; persuaded. A low use.

They are very sensible that they had better have pushed their conquests on the other side of the Adriatick; for then their territories would have lain together. Addison.
8. In low conversation
it has sometimes the sense of reasonable; judicious; wise.