To solve this attach a shackle or a shackle with some chain to the anchor first and then the swivel this will prevent the above mentioned problem with side loading.
Further it is often hard to size a swivel to match the strength of the HI TEST chain, though some well-designed products are available on the market they are often really expensive.
It is generally recommended to use a short pieces of chain attached to the anchor via a shackle and then attach the swivel bridging chain to chain. This will prevent both swivel jamming as seen above and the side loading issue.
There many brands available on the market and one should pay attention to the SWL (safe working load) rating on the swivel as compared to the chain.
Further if using a stainless steel swivel on the galvanized chain, try to isolate the two materials with insulating tape to reduce the galvanic interaction between the two dissimilar metals.