It is extremely unwise to run a chain which cannot comfortably shift into the big chainwheel/big cog combination; you risk derailleur or frame damage. Even if you do not intend to shift into that combination you may forget, and when you do it will be too late. Running a too-short chain is rarely a solution to shifting troubles but is a good way to create some.
Whether you can remove a link depends upon how much excess length you have.
As to cleaning/lubricating your derailleur, it couldn't hurt and may help; perhaps you should examine your cables (they may be dirty too), especially inside your brifters for fraying, and housings. New cables can often make a startling difference in shifting performance.