XPS is definitively not the right one, since it require ultra-high vacuum. If you suspect oxidation, then detect the oxidation products. For instance, functionalize them and do NMR or color reagents. They would also show up in GC or GC-MS after a silylation with e.g. BSTFA. Water could be there also, so check it with Karl Fischer titration. Finally, some sort of chromatography such as HPLC to estimate the distribution of molar masses would be helpful.