As a general remark ahead, I want to stress that this kind of question might not be considered on-topic on Stackoverflow, see How to ask. There are, however, related sites that might be better for these kinds of questions (no code, theoretical PoV), namely AI Stackexchange, or Cross Validated.
If you look at a rather popular paper in the field by Mueller and Thyagarajan, which is concerned with learning sentence similarity on LSTMs, they use a closely related dataset (the SICK dataset), which is also hosted by the SemEval competition, and ran alongside the STS benchmark in 2014.
Either one of those should be a reasonable set to fine-tune on, but STS has run over multiple years, so the amount of available training data might be larger.
As a great primer on the topic, I can also highly recommend the Medium article by Adrien Sieg (see here, which comes with an accompanied GitHub reference.
For semantic similarity, I would estimate that you are better of with fine-tuning (or training) a neural network, as most classical similarity measures you mentioned have a more prominent focus on the token similarity (and thus, syntactic similarity, although not even that necessarily). Semantic meaning, on the other hand, can sometimes differ wildly on a single word (maybe a negation, or the swapped sentence position of two words), which is difficult to interpret or evaluate with static methods.