I'm trying to make MOT tesla coil and would like to connect secondary coil in series. The low voltage end of secondary is connected to the body of the transformer. I don't particularly want to disconnect this as it may be useful for some other projects. Is it bad to connect the secondaries in series with secondary low end attached to transformer core but not connect the core to earth ground?
A MOT has one great attraction for amateur coilers, it's cheap.
Aside from that, it's the worst possible choice for powering Tesla Coils.
Let's assume you've done the obvious thing with two MOTs, which is to use one in phase, one in antiphase, to get a balanced +/- 2kV about ground, as Jasen has illustrated in his answer, with the primaries in parallel.
Voltage too Low
As you've found out, the voltage is still too low for reliable spark-gap operation. So, you're looking to stack them.
Here we need to make a distinction between the MOTs you used to be able to get 20 years ago, and those of the 2020s. Today's MOTs are value-engineered to the max. Old MOTs used to have thick insulation around both primary and secondary, which was good for the full output voltage, and then some. Today's MOTs only insulate the outside and sides of the secondary to the full HV output voltage, the inside of the secondary and the whole primary are only insulated to take mains spikes, 1500V or so.
That means although you may have seen posts from coilers saying they had lifted the core connection and put secondaries in series, or floated the core on the output of another MOT (and I've done both), that's because they were using old style over-engineered MOTs. This trick will not work as well, or at all, with today's MOTs.
There is a safe way to series two MOTs, if you have a third. Connect two MOTs together secondary to core, secondary to core. Now power a third MOT from the 'hot primary'. All insulation is now running within its design stress.
I may have the dots round the wrong way, I've not bothered to think it through. There are only two possibilities for phasing the connection of the hot primaries, one is right, the other will give zero output.
Let's say you use six MOTs to get the output of four in series, for a balanced +/- 4kV AC from ground. That's a lot of metal, and you're still barely at a voltage that gives adequate spark-gap results.
While a series connection will get you more volts when everything is working normally, sparks from a TC can get everywhere. A little extra voltage can break down the 3kV insulation. The first sparkover from your TC could trash your MOTs.
Voltage too High
The voltage is too high for safe 'on the bench' operation. I'm not saying that mains voltages are safe, but they're safe enough to use 'on the bench'. Below 1000V, any thickness of dry clothes will insulate against an accidental contact. A MOT secondary has a peak voltage of 3000V. That will jump through a layer of clothing.
Current too High
The current is far too high. Even if you leave the shunts in, they're not current-limiting like those of a Neon Sign Transformer (NST) are. If you get a shock from an NST, you're not guarranteed to be toast, that from a MOT is almost certainly fatal.
I feel your pain, NSTs are hard to come by. All the modern ones have GFCI or are high frequency, both of which renders them unsuitable for TC use. You need an old, second-hand, big-iron NST.
Join a dedicated coilers forum, google will find them for you. Get chatting to the old timers. Some will have NSTs that they're willing to sell or lend. It's sooooo much easier to start with the right tools, an NST, than to cobble inadequate and dangerous tools together.
A Good Way to make a Tesla Coil with a MOT
The one way a MOT does make a good TC is to balanced doubler rectify it to +/- 6kV DC, then double that through resonant charging into a rotary spark gap DC coil. It works very well, but it's rather complicated for a first project.
Well, it is dangerous because if you somehow come in contact with it, you will get a lethal shock... And considering the voltages, things like wood etc. start to become less reliable insulators. Doing this with MOTs is typically not recommended.
Also if you stack more then 1, you risk breakdown from the secondary to the primary.
it really depends on how good the insulation is between the primary and the core. these things are designed to be as cheap as is practical so it's probably not good enough.
can you work with a centre-grounded secondaty?
you can get that by useing two tranasformers with one of them reversed