I would suggest altering your bicycle position to take some of the weight of holding up your body off of your wrists. This is done by raising the handlebars while correspondingly moving your saddle back. This will most likely involve getting a shorter stem too. You might need to find a bicycle fitter that understands how to get a "gentleman's" position for comfort rather than how to make you go faster. Your goal is to find (by moving your saddle back just far enough) where your body is balanced over the pedals. When you get to that point, it magically takes the weight off of your hands. Of course moving your saddle back requires you to shorten your reach to the handlebars at the same time.

In general without knowing anymore about your present position, you probably will want your handlebars up high enough so it is level with your saddle. This most likely requires you to ride as big a frame as you can possibly straddle. It might involve installing a Nitto Technomic stem that lets you raise the bars higher. You will probably also need a seat post with greater setback. Nitto makes those too. Ordinary seat posts have 20/25mm of setback while A Velo Orange or Nitto has more.

Classic era bicycles - particularly Italian bicycles - were designed for racing with steeper seat and head angles. What you need is a bicycle that has only a 72 or even less seat angle. This allows your saddle to be further back. With the result that it takes your weight off of your handlebars. Touring bikes and some classic British bikes had swallower angles more suited to getting the kind of position you require.

It isn't just older buys taking my framebuilding class that find they prefer a further back bicycle position. Young urban riders that keep their eyes focused down the road for any potential danger prefer a more upright position too that results in slacker seat and head tube angles. Racing style of bikes don't convert well to a more upright position.