upgrading to synthetic oil in manual transmission of 2000- 626 V-6 , 140K miles OK?

2000 626 ES V-6
AS I said I have a 2000 626 ES V-6 with about 140,000 miles . Clutch has been replaced last fall . It is very hard to shift into 1st gear especially when cold . I have the clutch adjusted the best I can I think . If I adjust it in any more I start to get slippage when the pedal is all the way out, but I have to bury the pedal in the carpet to keep from grinding gears sometimes . I am wondering if the synchronizers might be wearing out . As far as I know the trans oil has never been changed . So I am thinking I should change it and am wondering if going to a synthetic oil might help with shifting . I have seen statements on the net that going to synthetic in an older tranny like mine can cause problems sometimes . Anybody have any thoughts on all this ?