Why won't my alero change gears to overdrive - 2003 Oldsmobile Alero

An overdrive basically is the same as another gear.
Driving in top gear and selecting overdrive will decrease the engine revs and maintain the same speed. The same as changing up a gear.

there is a code in the TCM that needs to be checked out.

There isn't an overdrive relay. There is an overdrive solenoid. It takes about four hours to replace. You better have manuals and patience to do it. otherwise you will mess up the transmission worse than it is already..I really don't recommend a nice to do it. When you take it apart, there are parts that fall out. You lose them or don't get them back in place, a transmission rebuild ,plus extra parts are in the works for you.

press the button in overdrive is on,button out overdrive is off,and wile accelerating if it goes all the way to the redline before changing gear overdrive is on!

Why will my 2003 Oldsmobile not shift into drive

first thing to check is transmission oil level and quality. If that is fine than your looking at either a valve body problem (valve body is what controls the selection of gears) or possibly a problem with the bands on the planetary gear set, they can be tightened in certain models. However i dont think it is the bands.
Does the overdrive only engage 4th gear? or does it feel like the gear ratios are different between overdrive and normal before the problem happened? some overdrive buttons engage final gear and some do as much as change gear ratios like in a cvt (contantly variable transmission) and some just change the timing in the gear changes to provide more rpm/power.
My best guess is that the valve controlling the overdrive in the valve body has failed and therefor you dont get a response from the transmission (i.e no gear shifting when overdrive is selected)

change shift solenoids

check vacuum hoses, change trans fluid and filter 

fourth clutch passage could be stop up fourth clutch seal could be leaking.3 and 4 shift valve could be stuck.

There isn't anything wrong with your car. This is perfectly normal. I have a 2000 ES300 as well as other people that I know and it operates in the same way. Overdrive doesn't mean the same thing for every car or transmission. In our case overdrive is simply a reference to the top gear which is 4th gear. When you switch overdrive off(which is depressing the button causing the O/D OFF dash light to illuminate,) the highest gear becomes 3rd because you turned 4th gear or overdrive off. The application scenario for this is usually mountainous terrain where a lot of uphill driving makes it more feasible not to up shift into 4th gear while ascending and descending. It will not switch back to overdrive or 4th gear until you re-engage it by turning the overdrive back on or repressing the button. Again, when the O/D OFF dash light is illuminated, overdrive which is 4th gear is off and the highest gear will then become 3rd. It will never shift into 4th gear while this light is illuminated or overdrive is off. When you press the button and the O/D OFF dash light is NOT illuminated overdrive is on and therefore it will shift into 4th gear as usual. This applies to all 97-2001 ES300's and probably any other car you drive tat exhibits this same type of change when overdrive is turned off.