Some clarification...The pattern calls for 29 sts and 32 rows? But we don't know what the basic gauge was for that unavailable yarn. Sometimes the gauge a pattern calls for may be very different than the label on the skein, especially for something like lace. The yarn you want to use says on the skein (or website or where ever you got the information) that the yarn normally gets 26 sts and 32 rows on the 4 (but I personally don't pay much attention to needle size, because not everyone will get the given gauge on the needles mentioned). And are you saying that you are actually getting 20 sts and 24 rows with the substitute yarn?
What I would be interested in knowing is the basic gauge and needle given for the original yarn called for. Comparing that basic gauge and needle size for that yarn and the one you want to substitute will tell you a bit about if the yarns are a similar weight to start with. If they are not you might have a hard time getting gauge and having a nice lacy effect.
If the pattern calls for 29 sts in 4 inches and you are getting only 20 you are going to come out a lot bigger. And yes, if you are using a smaller needle and still only getting that gauge it sounds like the lace will be much tighter. But if you went to a bigger needle you would be getting even bigger. It sounds like you need different yarn or will need to use way less stitches to make the shawl the same size given. With lace you usually use several sizes larger needle than normally called for for the yarn so that it is more open.
What you need to do is to get the gauge given if you want the shawl the size given (but with shawls exact size may not be that important). You also want to get a fabric that you are happy with. You might make a swatch of the yarn you want to use and see what you get after blocking. Do you like it? You could figure out mathematically from your swatch how big your shawl will turn out and if that seems within reason.