Micaiah, your 600 is a a 1984 model manufactured in November 1983. It is would have been a very low midrange model. I always find it amusing that Shogun spec'd Shimano 105 on a 600 model. The first generation 105 (.k.a. Golden Arrow) is a good midrange group of the era, but Shogun made some concessions. Only the three main tubes are CrMo and then they are the less expensive seamed tubes. The stays and forks are high tensile steel. The other notable concession, is the 27" wheelset, at a time when 700C was coming into vogue.

The fork definitely uses a 1" headset. Carbon forks with 1" steerer are plentiful and relatively inexpensive however you will have to cut the steerer and probably thread it. The other concern is that most carbon forks are made for 700C wheels. Consequentely, to use the OEM 27" wheelset, the brake pads have to raised about 4mm. It could be slightly more or less depending on the manufacturers fork dimensions (i.e. distance from droputs to brake mounting hole). Make sure you have the necessary adjustment range in the caliper slots.

As noted by BG, if that is the correct saddle height, the frame is far too big, unless you have a really long upper body, relative to your legs.