Do both sentences have the same meaning?
I was used to travelling alone, so having my whole family along has been a big adjustment for me to make.
I was used to travelling alone, so having my whole family along was a big adjustment for me to make.
(Could you please give more detailed explanation if i am wrong about something)
Let's look at your three options:
... has been a big adjustment...
This is the present, perfect continuous tense of the verb "to be". This tense indicates something that started in the past and continues to the present. So, in our case you started making the adjustment in the past and are still doing so today.
... was a big adjustment...
In this case we have the simple past. This tells us that at some point in the past a big adjustment was necessary. It tells us nothing about whether the adjustment is still necessary at present, or if you there was a process to continue the adjustment, simply that at one point in the past an adjustment was made. (To be clear, this gives us no information about the present or its continuity. It is perfectly possible that either you don't need to make the adjustment now, or you do. Either is compatible with the "was" since this sentence gives us no information about the present.)
... became a big adjustment...
You could use this idiomatically, but it is strictly incorrect. It tells us that "having the whole family along" transitioned to "being an adjustment". But that would imply that the "having" itself transitioned to the "adjustment", when in fact what you are trying to say is that the "having" caused the big adjustment. It is possible that there might be a figure of speech that would allow this, but it sounds like an odd thing to say.
In standard American English, the present perfect feels like a description of what is going on in the present based on something that happened in the past. The simple past simply describes something that happened in the past that may or may not have an ongoing effect on the present. It is possible that both sentences could be used to describe the same real-life situation if the adjustment happened in the recent past.
The first sentence, using "has been," leaves the listener thinking about the present and what the speaker still feels about the adjustment. The implication is that the adjustment was made "recently" and might undergo a further change. The adjustment is described as not yet being firm and requiring effort in the present.
The second sentence, using "was," leaves the listener thinking about the past and what the speaker might be remembering about this past event.
If the adjustment happened long ago, the second sentence is probably inappropriate. If it was recent, either could be used. In such a case, the first suggests that the speaker is still putting effort into preserving the adjustment or that the speaker still feels the effects of having to make the adjustment. Such ongoing effort or such an ongoing feeling justifies using what is essentially a present tense, rather than a past tense. Using the second would suggest that no more effort is necessary or that the feelings and the adjustment no longer requires current effort. This distinction is not required to be made in this case, but all speakers would feel the difference.
A speaker of standard American English would use the second sentence to narrate something that happened in the past. This tense would be appropriate if you were going on to explain aspects of the adjustment using more verbs in the simple past tense.
The first sentence would be used to explain the reasons for the speakers current feelings or concerns about the adjustment. If you continued to elaborate, you would likely use present-tense verbs to explain the ongoing situation.
