I suggest that unless someone here has worked on an ST180, you need to go through the process of familiarising yourself with it before you resort to the spanners.

Here is a place where you can download the service manual:
http://sonnys_bolens.tripod.com/manuals/suburban.htm

Under ST Suburban series press DOWNLOAD to get the 113 page service manual for the full ST series.

If you do some searching you will find that the ST180 is the advertising name for several different 3018 models. The 3018G had a Peerless 801-018C transmission, the 3018H-01 had an Eaton 7 hydrostatic drive/Peerless 1327C final drive combination, and the 3018H-02 had the Eaton 750. So, what transmission do you currently have?

Generically, to check a fault such as you have you begin by checking that the V belt driving the transmission is not slipping, the transmission shift linkage is engaging properly, the speed pedal is moving the external transmission control for its full stroke, and the transit lever has not been engaged. Then you inspect the transmission fluid level and condition. Often there is an externally-accessible filter, which may be blocked.

At that point, if the transmission is full of clean fluid but it does not drive forward, it looks as if there is an internal fault. Since it drives backwards, it will not be in the drive gears, but will be a blockage in the internal fluid filter (if it does not have an external one, it will have an internal one) or a fault in the internal hydro pump or hydro motor. You have to remove and dismantle the transmission to deal with any of those problems.

If the fluid is heavily discoloured it has been overheated. You may be lucky enough to get it working again with a complete fluid change and cleaned or replaced filter, but unless the abuse that cooked the fluid was brief and recent, pumping the damaged fluid through the transmission for a long period will have worn out the hydraulic pump, hydraulic motor, or both. Sometimes there are kits available to rebuild those parts.

And the very short summary is that there is a model of ST180 that used the same hydrostatic unit and final drive unit (Eaton 7/Peerless 1327C) that you have in your other ride-on, so it may or may not be practical to switch that unit into your ST180 if that is what you choose to do. If you tell us what transmission your ST180 currently has, we can talk about that.