First question is where are you experiencing the pins and needles? if it is primarily yor palm and your little and ring fingers it is likely to be caused by pressure on your ulnar nerve that runs through the centre of your palm. If it is more your palm and your thumb and forefinger then it could be that the problem is caused by carpal tunnel syndrome, caused by the position of your wrist in relation to the rest of your arm when you’re holding the bars. In either case, excessive vibration (and also the type/frequency of the vibration) is going t exacerbate the issue.
It’s important to understand the symptoms as fully as is possible, as the advice given would be different where it one or the other of the above (or indeed any other potential cause). It is probably worth seeing a bike experienced physio.
I had a similar problem until fairly recently. My main pub/commuting etc bike was an alu cross bike. Lovely bike, but my hands were killing me. Recently switched to a steel bike and the pain has gone. So it could be ride position (but the position is more aggressive on the new bike), handlebar tape (yet is thinner on the new bike), or a number of other things, but FWIW I think it’s actually that the steel frame and fork absorb vibration more efficiently and transfer way less of it to my hands.
That being the case I’d look at easy fixes that change the way in which road vibration is transmitted. Lots of good advice already on the thread, but if you don’t already have them I’d suggest carbon forks and handlebar, some really good tape and some decent gloves.
Whatever you do get it sorted though, as left to its own devices it’ll just get worse and worse.
I’ve always found Sheldon Brown to be a useful authority in most things bike. Not a huge amount on this, but worth a peruse nonetheless –
http://sheldonbrown.com/pain.html
EDIT – Interestingly, on the post linked to above, Sheldon puts forward the theory that too much padding – either on handlebars or gloves – can actually be detrimental and make things worse..