With the info you used to describe what is happening, I am not exactly sure what your problem is; blower not running while in the on position, or the compressor cycling off-on. If the blower motor is the problem, you need to replace that first. If the motor is in good shape and not pulling over 20 amps on high, you could have a bad module. That is what I described below.
The "blower control module", is located on top of the fan box under the fuse and relay section on the firewall. It has two wiring connectors connected to it and usually three screws holding it in the box. You can get a new one from any dealer for around $150.00. It is such a common problem on Buicks and other GM cars with auto fan control, that the dealers usually keep a good supply on hand. You can also check your local salvage yard, (about $50.00) they usually have some at the front counter. Be careful to insist on the same part number as the one you have removed if buying from a salvage yard. (The part number will be on the module). GM made four or five different controllers that look the same, however do not have the same number of wires or the same functions. You do not have a relay for the blower fan because the controller sends variable voltage to the fan to control the speed. Usually the resistors fail in the module and the default voltage is 12V direct to the blower, unswitched, therefore, your blower runs all the time unless you pull the fuse, disconnect the battery, or pull the plug on the fan itself. They also fail in the no run mode, or intermittent mode. The fuse is located on the firewall with the relays. Pull the cover off and look inside the cover to locate the correct one. It is a 30 amp fuse and labled "AC". I have also found that Advanced Auto Parts is now selling a new module for around $100.00, part # RU304 Standard products. It looks like this;
