Birch is heavier than ash. Birch 3/8" would be fine, but ash 3/8" will also not be a problem.
Try Plug It Dowels, the prices are excellent. His shafts aren't exactly premium arrow shafts, but they're so cheap you can buy in bulk and sort through to match them by rough spine. You can get ash, pine, ponderosa, birch and tulip poplar from him, in 1/2" and 3/8" sizes.
Type 10s are usually not allowed by most clubs, but you can get 3/8" brass / steel bullet points. I tend to buy the cheap Richard Head 11/32" ones and shove them onto 3/8" shafts with a bit of sanding to blend the join. I was shooting those out of a 110@30" bow all day yesterday with no problems.