Chemical engineers work in all areas of chemical processing not just the oil industry but they are more engineers than chemists.
You need to be looking at working for a chemical or pharmaceutical manufacturer.
I started work for a large chemical manufacturer in the laboratories more years ago than I care to remember!
First job was in the Process Control lab. The plant workers would bring me samples at different stages of the manufacturing process for me to analyse. I would pass the results to the Production Chemist who would decide whether the batch needed adjustment before continuing. Some was simple stuff like pH measurement, water content, viscosity or titration but in other cases I needed to do a lab scale conversion to the final product using different ratios of reactants to decide the reactant level for that batch.
From there I did R&D. Making lab scale preps to test a new source of raw material or looking for ways to improve the properties of existing products. My major project was determining the optimum chain length of various polyethoxylated alkyl phenols for use as wetting agents, detergents etc. AFter you have done a couple of hundred lab batches that gets quite tedious and I wanted to get into process chemistry - running a chemical plant!
Move of company as a shift production chemist controlling a plant manufacturing polyacrylamide copolymer flocculants. Biggest chemistry set I ever had!!!!! Based 50% in a lab doing testing of intermediates, raw materials, finished product and doing troubleshooting. Other 50% was out in the plant keeping an eye on things making sure all the processes were being followed.
From there I became lab manager - all the Quality Control. Raw Material testing etc. It was during this phase of my career that I realised I wasn't wearing a lab coat any more :(