Most Buicks these years have (2) battery cables on the positive side of the battery. It is extremely important that these cables are very clean where the battery cable bolts goes through them. If there is any corrosion in between these cable ends or under the battery bolts the starter will not work correctly. You must get these bolts out and clean the faces of the cables. The battery bolts will come out even if you have to trim the plastic around them to get them out. Replace these bolts if they are corroded and pitted. Any parts house stocks them. Usually the positive battery bolt is longer than the negative because it has to go through 2 cable ends. Clean the cable ends really well with a wire brush or sandpaper both insides top and bottom of cable connections until you can see shiny brass showing on the cable ends. I use a drill motor with a brush attachment. The problem is that a non-conductive residue that forms and insulates the cable end connections..
I can't stress enough how critically important this step is and is most times overlooked when a battery is replaced. After reconnecting the cables make sure the cables are completely tight and can't be rotated when you try to twist the cables.
As long as you can hear the starter trying to turn and is making a sound when you try to start I would rule out a problem with ignition switch. Also check and make sure the starter wire connections are tight and the battery has a full charge.

