A couple of things from me on this one.
1. Do not try putting iy in a chuck on either a pillar drill or in a lathe. A 3 jaw chuck is merely a holding device and is nowhere near accurate enough for this job.
2. If you have friend with a lathe get him to mount the shaft between centres and then simply rotate it by hand and use your DTI to check the run out.
3. If you cannot locate a lathe then get a piece of 6mm (or thicker) float glass from a local glass merchant. the thicker the better. Alternatively borrow a surface plate if you have a friend with one. them locate the shaft on a pair of Vee blocks, remembering that both blocks must locate on identical diameters on the shaft or you will not be able to measure anything. Then again use a DTI to check the run out.

so far so good, and now things to consider.
1. The diameters will most certainly have been sized by grinding after hardening.
2. the toleences will be very tight, probably in the low 10 thousandths of an inch...NOT ten thou (0.0100, but one ten thousandth of an inch (0.0001). So it begs the question about how you will get the shaft straightened to allow the sizes to match.
3. The shaft will have been hardened and tempered before final sizing on a grinder. there is a risk that pressing the shaft to straighten it might cause it to break.
4. Check the runout as described, and I suspect that it will still be true, and then consider that the movement that you have detected mat be due to bushes wearing, or after very high mileage the shaft may have worn.

I don't know where you are located Chaz but if you could get the shaft to me I will happily check it both for straightness and size/wear. I am in Leicestershire.