Widen your search for cordage
They're not all straight-down 90's (maybe in that store or vendor). You need any appliance cord that is of sufficiently large ampacity - e.g. 12 AWG is a little excessive for a cord with a NEMA 5-15 plug on it, but it assures legality/safety (nobody can argue with it.
The cordage must be SJOW, SJOOW or any of about a dozen types legal for flexible cord. The cordage must be marked. I generally have several sources for it.
- Appliance cords with wires or lugs on the other end, just like you're buying, but I widen my search out to places like McMaster-Carr, Galco, Grainger, Mouser, Digi-Key etc. I routinely buy straight-out 12 AWG cords from McMaster-Carr.
- A factory-built extension cord with molded-on socket and plug, marked on the cordage with correct gauge and type(SJOOW etc.) SNIP! goes the socket end, and now I have an appliance cord.
- Dead appliances, again of appropriate wire gauge and markings on the cordage itself.
- Bare Cordage bought from the electrical supply house of correct rating, and a field-attachable plug with appropriate strain relief for the cordage (i.e. Not random Home Depot plugs, which are sized for 16 AWG). This must be cordage, never house wiring such as NM, UF, AC, BX etc.
I am not a fan of field-attachable plugs anywhere near where water might be, unless the plug is protected by GFCI.
Must be handle-tied
Glad we caught you on this; an important part of Code requires that the 2 circuits be handle-tied since they are on the same yoke (receptacle). The handle-ties must be proper, UL listed items for that breaker, you can't use a nail (and the breaker must be UL listed or classified for that panel).
Handle-ties can be annoying to find, especially for double-stuff breakers, so we recommend using a 2-pole breaker instead simply because everyone sells them. The only difference between a 2-pole breaker and 2 handle-tied breakers is that a 2-pole breaker guarantees common trip. This doesn't require common trip, just common maintenance shuoff.
Since you wired independent neutrals, the two circuits are allowed to be on the same pole. However, it will be difficult to arrange them on the same pole with handle-ties, and impossible with 2-pole. (A 3-pole/phase breaker or tie would do, but then, you'd have an unrelated or dead circuit in the middle - yet another reason plenty of extra panel spaces are always a good idea.)
Breaker ampacity must exactly match socket ampacity
You know how normally, you can put the common NEMA 5-15 receptacles on 20A circuits just fine? Not here.
Because this feeds precisely one socket, the breaker must match the socket exactly. If your breakers are 20A, you must use a NEMA 5-20 receptacle, not 5-15.
You're not required to use cord-and-plug connection. You can hardwire one or both appliances. This also eliminates the handle-tie and receptacle-ampacity problems shown above.
A very easy way to get a port to run the wires into is to use a Surface Conduit Starter Kit.