Szenchyi is the busiest of the baths, and the most crowded, especially with tourists.
With 1.7 million annual visitors, 15 indoor pools and almost 3,000 sqm of water surface, it feels more like an amusement park than a thermal bath. (Source)
Early morning hours are less crowded, but since your traveling partner doesn't want to go at 6:00am (neither would I), that doesn't mean you should give up.
The pools are almost exclusively used by local regulars who wish to do their swimming laps in the outdoor sports pool between 6am and 8am, or who wish to use the spa facilities when they are not packed. Later on the day, especially mid and late afternoon, the baths and pools fill up with locals and tourists – especially in the summer months (from June to September).
So, if you want to pick the best time of the day, we recommend visiting the baths earlier in the day, either in the morning around 10am, or around noon to 1pm. As one of the tourists wrote in their Szechenyi Baths review on a travel forum “We were there today, arrived at 1.30pm, and by 3.30pm it was chock-full of tourists. Earlier it was quiet and only older locals were bathing.”
Another source says the evening hours are good too, because the tourists are off eating or visiting other landmarks.
There are various tickets, including discount bath tickets for early morning swimmers (from 6am to 8am), afternoon bath tickets (between 5pm and 7pm) and night bath tickets (outdoor pools only, between 7 pm and 9:45pm). Sometimes this encourages a crowd, but more often it's used for the local people who regularly frequent the baths.
There are a number of resources to guide you to the quieter spots. I really like Budapest by Locals. It gives the information from the perspective of people who live there, which is valuable when you don't want to go to a crowded, tourist-driven, bath.
Their Guide to Thermal Baths is an excellent comprehensive reference for all of the the major baths. Almost everything you need to know, including locations, facilities, hours of operation, prices, pictures, neighboring attractions, and other features, are nicely laid out. Pay close attention to prices and rules, because there are limitations depending on certain factors. There are also discounts you can use, so keep an eye out for those!
Next to Szecheny, the most popular are:
Gellert Bath. I recommend it to you because Monday is a great day to go there, and late morning into the mid-day is as good as early morning. It's as popular as Szecheny as a tourist destination, but is high-end, expensive and luxurious, with many services geared toward couples, but just as many for singles of friend groups. It also has a salt room, which others don't. From what I've read, that in itself is a worthwhile experience.
Kiraly Bath. This bath has a higher percentage of locals. One source says up to 50%.
Lukacs Bath offers a lot of the same services as the others, but is less touristy than the Gellért or the Széchenyi, if you prefer to mingle with locals than travelers.
Rudas Bath is a Turkish-style bath, originally designed for gender-segregated, often nude, bathing. In recent years, it has become more co-ed, with some days still reserved for male or female. Even on mixed days, required swimwear covers much less of the body than other baths. It's more of an adult-related experience, but it's very authentic.
At the far end of the spectrum as far as quiet is concerned, is Dandar. It was built as a city bath to service the local people. The general percentage of tourists is 10 percent. Until the last few years, they weren't even on the lists at some of the popular websites. They are trying to attract more tourists, though, by offering discounts for some admission and services. While smaller than some others, there's still plenty to do, and it's in a nice historical area.
Since you're going in March, you may want to note that it's a busy month in that region, with many festivals, holidays, fun activities for both tourists and locals. Budapest by locals' things to do in Budapest in March, Top Budapest's Things to do in March-Early Spring Events in Budapest are just a few resources to give you ideas. It's a combination of locals and tourists. Check your calendar for National Holidays, because a lot of things will be closed. Most of the baths stay open, though. March also has evening "Bath Parties," which are popular with locals and tourists. Check out online guides for more information. If you're only going to be in the area for one day, maybe it will coincide with a fun event!
I hope this reaches you in time for your trip, and gives you some options. When you get back, it would be great if you update your question or provide your own answer. Many of us, like me, have only read about these places, so the perspective of a recent visitor would be excellent!