I have done a separate DS160 form for me and my wife and paid the visa fee separately, so we have different appointment dates. In the DS 160 form I mentioned that she would be an accompanying traveler. Would that be an issue? Do we need to make a single appointment?
I don't know where you are from or what you think a spouse means. The fact that you had to mention you filled a "different DS-160 form" is a bit disturbing. And that you had to ask that if you can book separate appointments is a bit more.
You and your spouse are two different individuals. You HAVE to fill two different DS-160 forms and book two appointments. US visas (especially non-immigrant visas) are granted to individuals and assessed individually. The fact the other individual happens to be your spouse doesn't change anything in the application process.
You are two different people and will be assessed individually as per your applicantions and interview.
If you can get appointments on the same dates, it can be convenient to you as you live together. Nothing more, nothing less. If not, it is irrelevant to the US State department.
Do not worry about such irrelevant factors. Just follow what's given on the website. Two people, two passports, two forms, two appointments, two interviews, two visa evaluations. It is very much possible that only one of you gets the visa amd the other doesn't. That fact that you plan to travel together is ot relevant. It is quite common among families that some get a visa and some don't.
Everyone is an individual and is treated like one.