Loose Blade, Tight Arbor Bolt

If you're sure it's the shaft that's loose, and not a misalignment in the blade attachment, then it's most likely one of the bearings supporting the shaft that's bad. Check for play in the bearing flange (Component 36). If that's solid, then check the bearing sleeve (component 13). Unfortunately if it's the armature that's damaged (component 3), then you're better off buying a new saw.

Does the old blade have a diamond shaped hole? My 77 is a few miles away, but my recollection is that it uses that configuration. Some new blades won't have it at all (and thus are not suitable), others will have a round hole and a pre-cut diamond hole that punches out to engage the diamond shape on the saw clamps.

Here's the manual, which suggests both the "left hand thread" and diamond arbor as correct recollections on my part:

http://mdm.boschwebservices.com/files/SHD77M-02-73%20Manual%20r113863v5.pdf

Here is a blade which appears to have the diamond hole possiblity, but not yet knocked-out.

Here is one with the hole cleared to the diamond shape:

Note that the worm drive saw, while you prefer the feel/heft of it, can be rather more dangerous than a direct drive saw if it's jammed and kicks back. Most people treat them with a healthy dose of respect - or learn to eventually.